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+On vector-valued functions and the
+ε-product
+Habilitationsschrift
+vorgelegt am 31.01.2022
+der Technischen Universität Hamburg
+von
+Dr. rer. nat. Karsten Kruse,
+geboren am 19.11.1984 in Papenburg.
+Die Habilitationsschrift wurde in der Zeit von Juli 2020 bis Januar 2022 im
+Institut für Mathematik der Technischen Universität Hamburg angefertigt.
+arXiv:2301.13612v1  [math.FA]  31 Jan 2023
+
+Gutachter: Prof. José Bonet
+Prof. Dr. Leonhard Frerick
+Prof. Dr. Thomas Kalmes
+PD Dr. Christian Seifert
+eingereicht: 31. Januar 2022; überarbeitet: 30. Januar 2023
+Tag des Habilitationskolloquiums: 01. Juli 2022
+DOI: 10.15480/882.4898
+ORCID:
+0000-0003-1864-4915
+Creative Commons Lizenz:
+Diese Arbeit steht unter der Creative Commons Lizenz Namensnennung 4.0 (CC
+BY 4.0). Das bedeutet, dass sie vervielfältigt, verbreitet und öffentlich zugänglich
+gemacht werden darf, auch kommerziell, sofern dabei stets der Urheber, die Quelle
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+kann unter https://creativecommons.org/licenses/by/4.0/legalcode.de auf-
+gerufen werden.
+
+Acknowledgement
+It is quite hard to express how grateful I am to the people who helped, in one
+way or the other, to finish this thesis which spans a part of my work between 2016
+and 2022. But I will give it a try.
+First of all, I am deeply indebted to Marko Lindner and Christian Seifert who
+always supported and encouraged me since I joined the TUHH in 2014 and gave
+me a home so that I could work on the kind of mathematics I love. I know how
+lucky I was to meet you both.
+Second, I would like to thank the two people
+who taught me probably the most I know about complex analysis and functional
+analysis, namely, Andreas Defant and Michael Langenbruch (Oldenburg). Third, I
+am utterly grateful to José Bonet and Enrique Jordá (Valencia) for many helpful
+suggestions and comments, improving some of the papers this thesis is based on,
+as well as enduring the quite abstract setting.
+Let us come to the honorable mentions, ... just kidding. In 2015 I was lucky
+again because Jan Meichsner joined the TUHH as a PhD student. Despite him
+being a physicist and a dispraiser of green cabbage, it was a real pleasure to share
+an office, do mathematics or just spend time with him.
+Further, let me thank
+Dennis Gallaun with whom I spent a lot of effort and gaffer tape setting up the
+mobile e-assessment center at the TUHH between 2018 and 2020.
+Apart from the people mentioned above I would like to thank my other mathe-
+matical co-authors Hans Daduna, Ruslan Krenzler, Felix Schwenninger and Lin
+Xie whose work is not physically present in this thesis but whose mathematical
+influence or spirit probably is. Furthermore, I am thankful to the whole Institute
+of Mathematics of the TUHH, in particular, the group of Applied Analysis for their
+support. Moreover, I am grateful to the reviewers of this thesis Leonhard Frerick
+(Trier) and Thomas Kalmes (Chemnitz) besides José Bonet and Christian Seifert,
+and the anonymous reviewers of the papers it is based on for their work, helpful
+comments and corrections.
+Finally, I would like to thank my family for their continuous support and my
+love Sonja for sharing my mathematical interests, bearing my kind of humour and
+so much more which I cannot put into words.
+These words only roughly express my gratitude and I hope that the minimum
+that remains after reading the acknowledgement is the thought ‘At least, he gave
+it a try.’ and a smile.
+3
+
+
+Abstract
+This habilitation thesis centres on linearisation of vector-valued functions which
+means that vector-valued functions are represented by continuous linear operators.
+The first question we face is which vector-valued functions may be represented
+by continuous linear operators.
+We study this problem in the framework of ε-
+products and give sufficient conditions in Chapter 3 and 4 when a space F(Ω,E)
+of vector-valued functions on a set Ω coincides (up to an isomorphism) with the
+ε-product F(Ω)εE of a corresponding space of scalar-valued functions F(Ω) and
+the codomain E which is usually an infinite-dimensional locally convex Hausdorff
+space. The ε-product F(Ω)εE is a space of continuous linear operators from the
+dual space F(Ω)′ to E.
+Once we have a representation of a space F(Ω,E) of vector-valued functions
+by an ε-product F(Ω)εE, we have access to the rich theory of continuous linear
+operators which allows us to lift results that are known for the scalar-valued case
+to the vector-valued case. The whole Chapter 5, which spans more than half of this
+thesis, is dedicated to this lifting mechanism. But we should point out that this
+is not only about transferring results from the scalar-valued to the vector-valued
+case. The results in the vector-valued case encode additional information for the
+scalar-valued case as well, e.g. we may deduce from the solvability of a linear partial
+differential equation in the vector-valued case affirmative answers on the parameter
+dependence of solutions in the scalar-valued case (see Section 5.1).
+In Section 5.2 we give a unified approach to handle the problem of extending
+functions with values in E, which have weak extensions in F(Ω), to functions in the
+vector-valued counterpart F(Ω,E) of F(Ω). We present different extension the-
+orems depending on the topological properties of the spaces F(Ω) and E. These
+theorems also cover weak-strong principles. In particular, we study weak-strong
+principles for continuously partially differentiable functions of finite order in Sec-
+tion 5.3 and improve the well-known weak-strong principles of Grothendieck and
+Schwartz. We use our results on the extension of vector-valued functions to de-
+rive Blaschke’s convergence theorem for several spaces of vector-valued functions
+and Wolff’s theorem for the description of dual spaces of several function spaces
+F(Ω) in Section 5.4 and 5.5. Starting from the observation that every scalar-valued
+holomorphic function has a local power series expansion and that this is still true
+for holomorphic functions with values in E if E is locally complete, we develop a
+machinery which is based on linearisation and Schauder decomposition to transfer
+known series expansions from scalar-valued to vector-valued functions in Section
+5.6. Especially, we apply this machinery to derive Fourier expansions for E-valued
+Schwartz functions and C∞-smooth functions on Rd that are 2π-periodic in each
+variable. The last section of Chapter 5 is devoted to the representation of spaces
+F(Ω,E) of vector-valued functions by sequence spaces, which can be used to iden-
+tify the coefficient spaces of the series expansions from the preceding section, if one
+knows the coefficient space in the scalar-valued case. Furthermore, we give several
+new conditions on the Pettis-integrability of vector-valued functions in Appendix
+A.2, which are, for instance, needed for the Fourier expansions in Section 5.6.
+5
+
+
+Kurzfassung
+Im Mittelpunkt dieser Habilitationsschrift steht die Linearisierung vektorwer-
+tiger Funktionen, d. h. vektorwertige Funktionen sollen durch stetige lineare Opera-
+toren dargestellt werden. Die erste Frage, der man sich stellen muss, ist, welche
+vektorwertigen Funktionen durch stetige lineare Operatoren dargestellt werden kön-
+nen.
+Wir untersuchen dieses Problem im Rahmen von ε-Produkten und geben
+hinreichende Bedingungen in Kapitel 3 und 4 an, wann ein Raum F(Ω,E) von
+vektorwertigen Funktionen auf einer Menge Ω mit dem ε-Produkt F(Ω)εE eines
+entsprechenden Raums skalarwertiger Funktionen F(Ω) und des Wertebereichs E
+(bis auf Isomorphie) übereinstimmt.
+Hierbei ist E üblicherweise ein unendlich-
+dimensionaler lokalkonvexer Hausdorff Raum. Das ε-Produkt F(Ω)εE ist ein Raum
+stetiger linearer Operatoren, die vom Dualraum F(Ω)′ nach E abbilden.
+Sobald wir eine Darstellung eines Raums F(Ω,E) von vektorwertigen Funk-
+tionen durch ein ε-Produkt F(Ω)εE gewonnen haben, ist es uns möglich die reich-
+haltige Theorie der stetigen linearen Operatoren zu nutzen, die es uns erlaubt,
+Ergebnisse, die für den skalarwertigen Fall bekannt sind, auf den vektorwertigen
+Fall zu übertragen. Das gesamte Kapitel 5, das mehr als die Hälfte dieser Arbeit
+einnimmt, widmet sich diesem Übertragungsmechanismus.
+Es sei jedoch darauf
+hingewiesen, dass es hier nicht nur um die Übertragung von Ergebnissen aus dem
+skalarwertigen auf den vektorwertigen Fall geht. Die Ergebnisse im vektorwertigen
+Fall beinhalten auch zusätzliche Informationen für den skalarwertigen Fall, z. B.
+können wir aus der Lösbarkeit einer linearen partiellen Differentialgleichung im
+vektorwertigen Fall Antworten auf die Frage nach der Parameterabhängigkeit der
+Lösungen im skalarwertigen Fall ableiten (siehe Abschnitt 5.1).
+In Abschnitt 5.2 stellen wir einen einheitlichen Ansatz zur Lösung des Fort-
+setzungsproblems von Funktionen mit Werten in E, die schwache Fortsetzungen in
+F(Ω) haben, zu Funktionen im vektorwertigen Gegenstück F(Ω,E) von F(Ω) vor.
+Wir präsentieren verschiedene Fortsetzungssätze in Abhängigkeit von den topologi-
+schen Eigenschaften der Räume F(Ω) und E. Diese Sätze decken auch schwach-
+stark Prinzipien ab.
+Insbesondere untersuchen wir schwach-stark Prinzipien für
+endlich oft stetig partiell differenzierbare Funktionen in Abschnitt 5.3 und verbes-
+sern die bekannten schwach-starken Prinzipien von Grothendieck und Schwartz.
+Zudem leiten wir von unseren Ergebnissen zur Fortsetzung vektorwertiger Funktio-
+nen den Konvergenzsatz von Blaschke für diverse Räume vektorwertiger Funktionen
+ab und übertragen den Satz von Wolff auf Dualräume mehrerer Funktionenräume
+F(Ω) in den Abschnitten 5.4 und 5.5.
+Ausgehend von der Beobachtung, dass
+jede skalarwertige holomorphe Funktion eine lokale Potenzreihenentwicklung hat
+und dass dies auch für holomorphe Funktionen mit Werten in E gilt, wenn E
+lokal vollständig ist, entwickeln wir einen Mechanismus, der auf Linearisierung und
+Schauder-Zerlegung basiert, um in Abschnitt 5.6 bekannte Reihenentwicklungen
+von skalarwertigen auf vektorwertige Funktionen zu erweitern. Insbesondere wen-
+den wir diesen Mechanismus an, um Fourier-Entwicklungen für E-wertige Schwartz-
+Funktionen und C∞-glatte Funktionen auf Rd, die 2π-periodisch in jeder Variablen
+sind, zu erhalten.
+Der letzte Abschnitt von Kapitel 5 ist der Darstellung von
+7
+
+8
+KURZFASSUNG
+Räumen F(Ω,E) vektorwertiger Funktionen durch Folgenräume gewidmet, was
+man dazu nutzen kann, die Koeffizientenräume der Reihenentwicklungen aus dem
+vorangegangenen Abschnitt zu bestimmen, sofern man den Koeffizientenraum im
+skalarwertigen Fall kennt. Außerdem legen wir mehrere neue Bedingungen für die
+Pettis-Integrierbarkeit von vektorwertigen Funktionen in Anhang A.2 dar, die z. B.
+für die Fourier-Entwicklungen in Abschnitt 5.6 benötigt werden.
+
+Contents
+Acknowledgement
+3
+Abstract
+5
+Kurzfassung
+7
+Chapter 1.
+Introduction
+11
+Chapter 2.
+Notation and preliminaries
+17
+Chapter 3.
+The ε-product for weighted function spaces
+21
+3.1.
+ε-into-compatibility
+21
+3.2.
+ε-compatibility
+28
+Chapter 4.
+Consistency
+37
+4.1.
+The spaces AP(Ω,E) and consistency
+37
+4.2.
+Further examples of ε-products
+41
+4.3.
+Riesz–Markov–Kakutani representation theorems
+59
+Chapter 5.
+Applications
+67
+5.1.
+Lifting the properties of maps from the scalar-valued case
+67
+5.2.
+Extension of vector-valued functions
+74
+5.2.1.
+Extension from thin sets
+76
+5.2.2.
+Extension from thick sets
+92
+5.3.
+Weak-strong principles for differentiability of finite order
+104
+5.4.
+Vector-valued Blaschke theorems
+110
+5.5.
+Wolff type results
+114
+5.6.
+Series representation of vector-valued functions via Schauder
+decompositions
+117
+5.6.1.
+Schauder decomposition
+120
+5.6.2.
+Examples of Schauder decompositions
+123
+5.7.
+Representation by sequence spaces
+130
+Appendices
+Appendix A.
+Compactness of closed absolutely convex hulls and Pettis-
+integrals
+139
+A.1.
+Compactness of closed absolutely convex hulls
+139
+A.2.
+The Pettis-integral
+142
+List of Symbols
+147
+Index
+151
+Bibliography
+153
+9
+
+
+CHAPTER 1
+Introduction
+This work is dedicated to a classical topic, namely, the linearisation of weighted
+spaces of vector-valued functions. The setting we are interested in is the following.
+Let F(Ω) be a locally convex Hausdorff space of functions from a non-empty set Ω
+to a field K and E be a locally convex Hausdorff space over K. The ε-product of
+F(Ω) and E is defined as the space of linear continuous operators
+F(Ω)εE ∶= Le(F(Ω)′
+κ,E)
+equipped with the topology of uniform convergence on equicontinuous subsets of
+the dual F(Ω)′ which itself is equipped with the topology of uniform convergence
+on absolutely convex compact subsets of F(Ω). Suppose that the point-evaluation
+functionals δx, x ∈ Ω, belong to F(Ω)′ and that there is a locally convex Hausdorff
+space F(Ω,E) of E-valued functions on Ω such that the map
+S∶F(Ω)εE → F(Ω,E), u �→ [x ↦ u(δx)],
+(1)
+is well-defined. The main question we want to answer reads as follows. When is
+F(Ω)εE a linearisation of F(Ω,E), i.e. when is S an isomorphism?
+In [15, 16, 17] Bierstedt treats the space CV(Ω,E) of continuous functions on
+a completely regular Hausdorff space Ω weighted with a Nachbin-family V and its
+topological subspace CV0(Ω,E) of functions that vanish at infinity in the weighted
+topology. He derives sufficient conditions on Ω, V and E such that the answer
+to our question is affirmative, i.e. S is an isomorphism.
+Schwartz answers this
+question for several weighted spaces of k-times continuously partially differentiable
+functions on Ω = Rd like the Schwartz space in [158, 159] for quasi-complete E
+with regard to vector-valued distributions.
+Grothendieck treats the question in
+[83], mainly for nuclear F(Ω) and complete E. In [99, 100, 101] Komatsu gives a
+positive answer for ultradifferentiable functions of Beurling or Roumieu type and
+sequentially complete E with regard to vector-valued ultradistributions. For the
+space of k-times continuously partially differentiable functions on open subsets Ω
+of infinite dimensional spaces equipped with the topology of uniform convergence
+of all partial derivatives up to order k on compact subsets of Ω sufficient conditions
+for an affirmative answer are deduced by Meise in [129]. For holomorphic functions
+on open subsets of infinite dimensional spaces a positive answer is given in [52]
+by Dineen. Bonet, Frerick and Jordá show in [30] that S is an isomorphism for
+certain closed subsheaves of the sheaf C∞(Ω,E) of smooth functions on an open
+subset Ω ⊂ Rd with the topology of uniform convergence of all partial derivatives
+on compact subsets of Ω and locally complete E which, in particular, covers the
+spaces of harmonic and holomorphic functions.
+An important application of linearisation is within the field of partial differen-
+tial equations. Let E be a linear space of functions on a set U and P(∂)∶C∞(Ω) →
+C∞(Ω) a linear partial differential operator with C∞-smooth coefficients where
+C∞(Ω) ∶= C∞(Ω,K). We call the elements of U parameters and say that a family
+(fλ)λ∈U in C∞(Ω) depends on a parameter w.r.t. E if the map λ ↦ fλ(x) is an
+element of E for every x ∈ Ω. The question of parameter dependence is whether for
+11
+
+12
+1. INTRODUCTION
+every family (fλ)λ∈U in C∞(Ω) depending on a parameter w.r.t. E there is a family
+(uλ)λ∈U in C∞(Ω) with the same kind of parameter dependence which solves the
+partial differential equation
+P(∂)uλ = fλ,
+λ ∈ U.
+In particular, it is the question of Ck-smooth (holomorphic, distributional, etc.)
+parameter dependence if E is the space Ck(U) of k-times continuously partially
+differentiable functions on an open set U ⊂ Rd (the space O(U) of holomorphic
+functions on an open set U ⊂ C, the space of distributions D(V )′ on an open set
+V ⊂ Rd where U = D(V ), etc.). The question of parameter dependence w.r.t. E has
+an affirmative answer for several locally convex Hausdorff spaces E due to tensor
+product techniques and splitting theory. Indeed, the answer is affirmative if the
+topology of E is stronger than the topology of pointwise convergence on U and
+P(∂)E∶C∞(Ω,E) → C∞(Ω,E)
+is surjective where P(∂)E is the version of P(∂) for E-valued functions. The oper-
+ator P(∂)E is surjective if its version P(∂) for scalar-valued functions is surjective,
+for instance, if P(∂) is elliptic, and E is a Fréchet space. This is a consequence
+of Grothendieck’s theory of tensor products [83], the nuclearity of C∞(Ω) and the
+isomorphism C∞(Ω,E) ≅ C∞(Ω)εE for locally complete E. Thanks to the splitting
+theory of Vogt for Fréchet spaces [173] and of Bonet and Domański for PLS-spaces
+[54] we even have in case of an elliptic P(∂) that P(∂)E for d > 1 is surjective if
+E ∶= F ′
+b where F is a Fréchet space satisfying the condition (DN) or if E is an ultra-
+bornological PLS-space having the property (PA) since kerP(∂) has the property
+(Ω). In particular, these three results cover the cases that E = Ck(U), O(U) or
+D(V )′. Of course, this technique to answer the question of parameter dependence
+is not restricted to linear partial differential operators or the space C∞(Ω).
+Another application of linearisation lies in the problem of extending a vector-
+valued function f∶Λ → E from a subset Λ ⊂ Ω to a locally convex Hausdorff space
+E if the scalar-valued functions e′ ○ f are extendable for each continuous linear
+functional e′ from certain linear subspaces G of E′ under the constraint of preserving
+the properties, like holomorphy, of the scalar-valued extensions. This problem was
+considered, among others, by Grothendieck [82, 83], Bierstedt [17], Gramsch [77],
+Grosse-Erdmann [79, 81], Arendt and Nikolski [6, 7, 8], Bonet, Frerick, Jordá and
+Wengenroth [30, 69, 70, 92, 93]. Even the simple case Λ = Ω and G = E′ is interesting
+and an affirmative answer is called a weak-strong principle.
+Our goal is to give a unified and flexible approach to linearisation which is able
+to handle new examples and covers the already known examples.
+Organisation of this thesis
+After fixing some notions and preliminaries on locally convex Hausdorff spaces,
+continuous linear operators and continuously partially differentiable functions in
+Chapter 2, we study the problem of linearisation in Chapter 3. In Section 3.1 we
+introduce our standard example of spaces F(Ω,E) that we consider. Namely, spaces
+of functions FV(Ω,E) from Ω to E which are subspaces of sections of domains of
+linear operators T E on EΩ, and whose topology is generated by a family of weight
+functions V. These spaces cover many examples of classical spaces of functions
+appearing in analysis like the mentioned ones and an example of the operators T E
+are the partial derivative operators. Then we exploit the structure of our spaces
+to describe a sufficient condition, which we call consistency, on the interplay of the
+pairs of operators (T E,T K) and the map S such that S becomes an isomorphism
+into, i.e. an isomorphism to its range (see Theorem 3.1.12).
+
+ORGANISATION OF THIS THESIS
+13
+In Section 3.2 we tackle the problem of surjectivity of S. In our main Theorem
+3.2.4 and its Corollary 3.2.5 we give several sufficient conditions on the pairs of
+operators (T E,T K) and the spaces involved such that S∶FV(Ω)εE → FV(Ω,E) is
+an isomorphism. Looking at the pair of partial differential operators (P(∂)E,P(∂))
+considered above, these conditions allow us to express P(∂)E as P(∂)E = S ○
+(P(∂)εidE) ○ S−1 where P(∂)εidE is the ε-product of P(∂) and the identity idE
+on E. Hence it becomes obvious that the surjectivity of P(∂)E is equivalent to
+the surjectivity of P(∂)εidE. This is used in [105, 109, 112, 116, 119] in the case
+of the Cauchy–Riemann operator P(∂) = ∂ on spaces of smooth functions with
+exponential growth.
+In Chapter 4 we take a closer look at the notion of consistency of (T E,T K). In
+Section 4.1 we characterise several properties of the functions S(u) for u ∈ FV(Ω)εE
+that are inherited from the elements of FV(Ω).
+Section 4.2 is devoted to several concrete examples of spaces of vector-valued
+functions that may be linearised by S and which we use for our applications in the
+forthcoming sections and chapters.
+In Section 4.3 we answer in several cases the question whether given a con-
+tinuous linear functional T K on F(Ω) there is always a continuous linear operator
+T E on F(Ω,E) such that (T E,T K) is consistent. This is closely related to Riesz–
+Markov–Kakutani theorems for T K, which we transfer to the vector-valued case.
+Chapter 5 is dedicated to applications of linearisation. In Section 5.1 we come
+back to our problem of parameter dependence. We show in our main Theorem 5.1.2
+of this section how to use linearisations to transfer properties like injectivity, sur-
+jectivity or bijectivity from a map T K∶F1(Ω1) → F2(Ω2) to the corresponding map
+T E∶F2(Ω1,E) → F2(Ω2,E) if the pair (T E,T K) is consistent under suitable as-
+sumptions on the spaces involved. Besides the problem of parameter dependence
+for (hypo)elliptic linear partial differential operators (see Corollary 5.1.3), we de-
+duce a vector-valued version of the Borel–Ritt theorem (see Theorem 5.1.4) from
+this main theorem and give sufficient conditions under which the Fourier transfor-
+mation FC∶Sµ(Rd) → Sµ(Rd) on the Beurling–Björck space is still an isomorphism
+in the vector-valued case and may be decomposed as FE = S ○ (FCεidE) ○ S−1 (see
+Theorem 5.1.5).
+In Section 5.2 we present a general approach to the extension problem consid-
+ered above for a large class of function spaces F(Ω,E) if the map S is an isomor-
+phism into. The spaces we treat are of the kind that F(Ω) belongs to the class of
+semi-Montel, Fréchet–Schwartz or Banach spaces, or that E is a semi-Montel space.
+Apart from linearisation and consistency, the main ingredient of this approach is to
+view the set Λ ⊂ Ω from which we want to extend our functions as a set of function-
+als {δx ∣ x ∈ Λ}. This view allows us to generalise the extension problem in Question
+5.2.1 by swapping this set of functionals by other functionals, which opens up new
+possibilities in applications that we explore in Section 5.3, Section 5.4, Section 5.5
+and Section 5.7. In the extension problem we always have to balance the sets Λ
+from which we extend our functions and the subspaces G ⊂ E′ with which we test.
+The case of ‘thin’ sets Λ and ‘thick’ subspaces G is handled in Section 5.2.1 with
+main theorems Theorem 5.2.15, Theorem 5.2.20 and Theorem 5.2.29, the converse
+case of ‘thick’ sets Λ and ‘thin’ subspaces G is handled in Section 5.2.2 with main
+theorems Theorem 5.2.52, Theorem 5.2.63 and Theorem 5.2.69.
+In Section 5.3 we consider weak-strong principles for continuously partially
+differentiable functions of finite order. For locally complete E it is well-known that
+a function f belongs to C∞(Ω,E) if and only if e′ ○ f ∈ C∞(Ω) for all e′ ∈ E′ (see
+e.g. [30, Theorem 9, p. 232]). If k ∈ N0, then it is still true that f ∈ Ck(Ω,E) implies
+e′ ○ f ∈ Ck(Ω) for all e′ ∈ E′. But the converse is not true anymore. Only a weaker
+
+14
+1. INTRODUCTION
+version of this weak-strong principle holds which is due to Grothendieck [82] and
+Schwartz [158] (see Theorem 5.3.2). Namely, if k ∈ N0, E is sequentially complete
+and f∶Ω → E is such that e′ ○ f ∈ Ck+1(Ω) for all e′ ∈ E′, then f ∈ Ck(Ω,E). Using
+the results from Section 5.2, we improve this weaker version of the weak-strong
+principle by allowing E to be locally complete, only testing with less functionals
+from certain linear subspaces G ⊂ E′ and getting that f does not only belong to
+Ck(Ω,E) but that all partial derivatives of order k are actually locally Lipschitz
+continuous (see Corollary 5.3.5). If we restrict to semi-Montel spaces E, then even
+a ‘full’ weak-strong principle Theorem 5.3.6 holds as in the C∞-case.
+In Section 5.4 we derive vector-valued Blaschke theorems like Corollary 5.4.2
+for several function spaces.
+This generalises results of Arendt and Nikolski [7]
+for bounded holomorphic functions and Frerick, Jordá and Wengenroth [70] for
+bounded functions in the kernel of a hypoelliptic linear partial differential operator.
+These are results of the form: given a bounded net (fι)ι∈I in some space F1(Ω,E)
+of Banach-valued functions which converges pointwise on a certain subset of Ω there
+is a limit f ∈ F1(Ω,E) of this net w.r.t. a weaker topology of a linear superspace
+F2(Ω,E) of F1(Ω,E). In Blaschke’s classical convergence theorem [38, Theorem
+7.4, p. 219] we have E = C, F1(Ω,E) is the space of bounded holomorphic functions
+on the open unit disc D ⊂ C, F2(Ω,E) is the space of holomorphic functions on D
+and the weaker topology is the topology of compact convergence.
+In Section 5.5 we present Wolff type descriptions of the dual space of several
+function spaces F(Ω) using linearisation (see Theorem 5.5.1). Wolff’s theorem [183,
+p. 1327] (cf. [81, Theorem (Wolff), p. 402]) phrased in a functional analytic way
+(see [70, p. 240]) says: if Ω ⊂ C is a domain, then for each µ ∈ O(Ω)′ there are a
+sequence (zn)n∈N which is relatively compact in Ω and a sequence (an)n∈N in the
+space ℓ1 of absolutely summable sequences such that µ = ∑∞
+n=1 anδzn.
+In Section 5.6 we derive a general result for Schauder decompositions of the
+ε-product FεE for locally convex Hausdorff spaces F and E if F has an equicon-
+tinuous Schauder basis (see Theorem 5.6.1). In combination with linearisation and
+consistency this can be used for F = F(Ω) to lift series representations like the
+power series expansion of holomorphic functions from scalar-valued functions to
+vector-valued functions (see Corollary 5.6.5). We present several examples in Sec-
+tion 5.6.2, for instance, Fourier expansions in the Schwartz space S(Rd,E) and in
+the space C∞
+2π(Rd,E) of functions in C∞(Rd,E) that are 2π-periodic in each vari-
+able. In particular, we combine these expansions for locally complete E with the
+results from Section 5.1 to identify the coefficient spaces of the Fourier expansions
+in S(Rd,E) and C∞
+2π(Rd,E) (see Theorem 5.6.13 and Theorem 5.6.14).
+In Section 5.7 an application of our extension results from Section 5.2 is given to
+represent function spaces F(Ω,E) by sequence spaces if one knows such a represen-
+tation for F(Ω) (see Theorem 5.7.1). As examples we treat the space O(DR(0),E)
+of E-valued holomorphic functions on the disc DR(0) ⊂ C around 0 with radius
+0 < R ≤ ∞ and the multiplier space OM(R,E) of the Schwartz space for locally
+complete E (see Corollary 5.7.2, Corollary 5.7.3 and Remark 5.7.4).
+The first section Appendix A.1 of the Appendix A is devoted to the question
+when the closure of an absolutely convex hull of a set is compact in a locally convex
+Hausdorff space E and Appendix A.2 to the related question of Pettis-integrability
+of an E-valued function.
+
+CONCERNING ORIGINALITY
+15
+Concerning originality
+We note that some parts of chapters or sections are based on our papers and
+preprints.
+● Chapter 3, Section 4.1 and Section 4.2 are based on our paper Weighted
+spaces of vector-valued functions and the ε-product [110] and its extended
+preprint [106]. Furthermore, Section 4.2 contains results from Sections 3
+and 6 of our accepted preprint Extension of weighted vector-valued func-
+tions and sequence space representation [115] and our paper Extension of
+weighted vector-valued functions and weak–strong principles for differen-
+tiable functions of finite order [117] and its extended preprint [120].
+● Section 5.1 generalises some results of our papers Surjectivity of the ∂-
+operator between weighted spaces of smooth vector-valued functions [116]
+and Parameter dependence of solutions of the Cauchy–Riemann equation
+on weighted spaces of smooth functions [112] and its extended preprint
+[108].
+● Section 5.2, Section 5.3, Section 5.4, Section 5.5 and Section 5.7 are based
+on our accepted preprint [115] and our paper [117] (and its extended
+preprint [120]).
+● Section 5.6 is based on our paper Series representations in spaces of vector-
+valued functions via Schauder decompositions [114].
+Moreover, the introduction Chapter 1 and Chapter 2 on notation and pre-
+liminaries are based on the corresponding sections in our papers and preprints
+[106, 110, 112, 114, 115, 116, 117, 120]. However, not all of the results given in this
+thesis are already contained in our preprints or papers.
+In Chapter 3 the new, i.e. not contained in our preprints or papers, results are
+Corollary 3.2.5 (ii), Example 3.2.7 e)+f), Example 3.2.9 and Corollary 3.2.10.
+In Section 4.2 the new examples and results are Example 4.2.2, Corollary
+4.2.3, Example 4.2.11, Example 4.2.13, Proposition 4.2.14, Example 4.2.16, Ex-
+ample 4.2.22 which extends [107, Proposition 3.17 a), p. 244] of our paper The
+approximation property for weighted spaces of differentiable function [107], Propo-
+sition 4.2.25 which extends [114, Proposition 4.8, p. 370] from sequentially complete
+E to locally complete E, Example 4.2.26 and Example 4.2.28 (ii). All the results
+of Section 4.3 are new except for Definition 4.3.1 which is [115, 2.2 Definition, p. 4]
+(and also not a result).
+The main theorem of Section 5.1, Theorem 5.1.2, is new even though special
+cases appeared in [112, 116]. Theorem 5.1.4 and Theorem 5.1.5 are new as well.
+Corollary 5.4.3 extends [120, 7.3 Corollary, p. 22] from metric spaces with finite
+diameter to arbitrary metric spaces. Theorem 5.6.13 and Theorem 5.6.14 b) extend
+[114, Theorem 4.9, p. 371–372] and [114, Theorem 4.11, p. 375] from sequentially
+complete E to locally complete E. Corollary 5.7.2 is new in the sense that there is
+only a sketch how to prove it in [115, p. 31].
+The results of Appendix A are also new except for Proposition A.1.1, Propo-
+sition A.1.6, which are contained in [106, 5.2 Proposition, p. 24] and [106, 3.13
+Lemma d), p. 10], and Lemma A.2.2 which is [114, Lemma 4.7, p. 369].
+
+
+CHAPTER 2
+Notation and preliminaries
+Basics of topology
+We equip the spaces Rd, d ∈ N, and C with the usual Euclidean norm ∣ ⋅ ∣,
+denote by Br(x) ∶= {w ∈ Rd ∣ ∣w − x∣ < r} the ball around x ∈ Rd and by Dr(z) ∶=
+{w ∈ C ∣ ∣w − z∣ < r} the disc around z ∈ C with radius r > 0. Furthermore, for a
+subset M of a topological space (X,t) we denote the closure of M by M and the
+boundary of M by ∂M. If we want to emphasize that we take the closure in X
+resp. w.r.t. the topology t, then we write M
+X resp. M
+t. For a subset M of a vector
+space X we denote by ch(M) the circled hull, by cx(M) the convex hull and by
+acx(M) the absolutely convex hull of M. If X is a topological vector space, we
+write acx(M) for the closure of acx(M) in X.
+Locally convex Hausdorff spaces and continuous linear operators
+By E we always denote a non-trivial, i.e. E ≠ {0}, locally convex Hausdorff
+space over the field K = R or C equipped with a directed fundamental system of
+seminorms (pα)α∈A and, in short, we write that E is an lcHs. If E = K, then we set
+(pα)α∈A ∶= {∣ ⋅ ∣}.
+By XΩ we denote the set of maps from a non-empty set Ω to a non-empty
+set X, by χK we mean the characteristic function of K ⊂ Ω, by C(Ω,X) the space
+of continuous functions from a topological space Ω to a topological space X, and
+by C0(Ω,X) its subspace of continuous functions that vanish at infinity if X is a
+locally convex Hausdorff space.
+We denote by L(F,E) the space of continuous linear operators from F to E
+where F and E are locally convex Hausdorff spaces.
+If E = K, we just write
+F ′ ∶= L(F,K) for the dual space and G○ for the polar set of G ⊂ F. If F and E
+are linearly topologically isomorphic, we just write that F and E are isomorphic,
+in symbols F ≅ E. We denote by Lt(F,E) the space L(F,E) equipped with the
+locally convex topology t of uniform convergence on the finite subsets of F if t = σ,
+on the absolutely convex, compact subsets of F if t = κ, on the absolutely convex,
+σ(F,F ′)-compact subsets of F if t = τ, on the precompact (totally bounded) subsets
+of F if t = γ and on the bounded subsets of F if t = b. We use the symbols t(F ′,F)
+for the corresponding topology on F ′ and t(F) for the corresponding bornology on
+F. We say that a subspace G ⊂ F ′ is separating (the points of F) if for every x ∈ F
+it follows from y(x) = 0 for all y ∈ G that x = 0. Clearly, this is equivalent to G
+being σ(F ′,F)-dense in F ′. For details and notions on the theory of locally convex
+spaces not explained in this thesis see [68, 89, 131, 138].
+ε-products and tensor products
+The so-called ε-product of Schwartz is defined by
+FεE ∶= Le(F ′
+κ,E)
+(2)
+where L(F ′
+κ,E) is equipped with the topology of uniform convergence on equicon-
+tinuous subsets of F ′. This definition of the ε-product coincides with the original
+17
+
+18
+2. NOTATION AND PRELIMINARIES
+one by Schwartz [159, Chap. I, §1, Définition, p. 18]. It is symmetric which means
+that FεE ≅ EεF. In the literature the definition of the ε-product is sometimes done
+the other way around, i.e. EεF is defined by the right-hand side of (2) but due to the
+symmetry these definitions are equivalent and for our purpose the given definition
+is more suitable. If we replace F ′
+κ by F ′
+γ, we obtain Grothendieck’s definition of the
+ε-product and we remark that the two ε-products coincide if F is quasi-complete
+because then F ′
+γ = F ′
+κ holds. However, we stick to Schwartz’ definition.
+For locally convex Hausdorff spaces Fi, Ei and Ti ∈ L(Fi,Ei), i = 1,2, we define
+the ε-product T1εT2 ∈ L(F1εF2,E1εE2) of the operators T1 and T2 by
+(T1εT2)(u) ∶= T2 ○ u ○ T t
+1,
+u ∈ F1εF2,
+where T t
+1∶E′
+1 → F ′
+1, e′ ↦ e′ ○ T1, is the dual map of T1. If T1 is an isomorphism
+and F2 = E2, then T1εidE2 is also an isomorphism with inverse T −1
+1 εidE2 by [159,
+Chap. I, §1, Proposition 1, p. 20] (or [89, 16.2.1 Proposition, p. 347] if the Fi are
+complete).
+As usual we consider the tensor product F ⊗E as a linear subspace of FεE for
+two locally convex Hausdorff spaces F and E by means of the linear injection
+Θ∶F ⊗ E → FεE,
+k
+∑
+n=1
+fn ⊗ en �→ [y ↦
+k
+∑
+n=1
+y(fn)en].
+(3)
+Via Θ the space F ⊗ E is identified with the space of operators with finite rank
+in FεE and a locally convex topology is induced on F ⊗ E. We write F ⊗ε E for
+F ⊗ E equipped with this topology and F ̂⊗εE for the completion of the injective
+tensor product F ⊗εE. For more information on the theory of ε-products and tensor
+products see [49, 89, 94].
+Several degrees of completeness
+The sufficient conditions for surjectivity of the map S∶F(Ω)εE → F(���,E)
+from the introduction, which we derive in the forthcoming, depend on assumptions
+on different types of completeness of E. For this purpose we recapitulate some
+definitions which are connected to completeness. We start with local completeness.
+For a disk D ⊂ E, i.e. a bounded, absolutely convex set, the linear space ED ∶=
+⋃n∈N nD becomes a normed space if it is equipped with the gauge functional of
+D as a norm (see [89, p. 151]). The space E is called locally complete if ED is a
+Banach space for every closed disk D ⊂ E (see [89, 10.2.1 Proposition, p. 197]). We
+call a non-empty subset A of an lcHs E locally closed if every local limit point of
+A belongs to A. Here, a point x ∈ E is called a local limit point of A if there is a
+sequence (xn)n∈N in A that converges locally to x (see [138, Definition 5.1.14, p.
+154–155]), i.e. there is a disk D ⊂ E such that (xn) converges to x in ED (see [138,
+Definition 5.1.1, p. 151]). The local closure of a subset A of E is defined as the
+smallest locally closed subset of E which contains A (see [138, Definition 5.1.18, p.
+155]). Moreover, we note that every locally complete linear subspace of E is locally
+closed and a locally closed linear subspace of a locally complete space is locally
+complete by [138, Proposition 5.1.20 (i), p. 155].
+Moreover, a locally convex Hausdorff space is locally complete if and only if it
+is convenient by [104, 2.14 Theorem, p. 20]. In particular, every complete locally
+convex Hausdorff space is quasi-complete, every quasi-complete space is sequentially
+complete and every sequentially complete space is locally complete and all these
+implications are strict. The first two by [89, p. 58] and the third by [138, 5.1.8
+Corollary, p. 153] and [138, 5.1.12 Example, p. 154].
+Now, let us recall the following definition from [181, 9-2-8 Definition, p. 134] and
+[175, p. 259]. A locally convex Hausdorff space is said to have the [metric] convex
+compactness property ([metric] ccp) if the closure of the absolutely convex hull of
+
+VECTOR-VALUED CONTINUOUSLY PARTIALLY DIFFERENTIABLE FUNCTIONS
+19
+every [metrisable] compact set is compact. Sometimes this condition is phrased
+with the term convex hull instead of absolutely convex hull but these definitions
+coincide. Indeed, the first definition implies the second since every convex hull of
+a set A ⊂ E is contained in its absolutely convex hull. On the other hand, we have
+acx(A) = cx(ch(A)) by [89, 6.1.4 Proposition, p. 103] and the circled hull ch(A) of a
+[metrisable] compact set A is compact by [153, Chap. I, 5.2, p. 26] [and metrisable
+by [34, Chap. IX, §2.10, Proposition 17, p. 159] since D × A is metrisable and
+ch(A) = ME(D × A) where ME∶K × E → E is the continuous scalar multiplication
+and D ∶= D1(0) the open unit disc], which yields the other implication.
+In particular, every locally convex Hausdorff space with ccp has obviously met-
+ric ccp, every quasi-complete locally convex Hausdorff space has ccp by [181, 9-2-10
+Example, p. 134], every sequentially complete locally convex Hausdorff space has
+metric ccp by [23, A.1.7 Proposition (ii), p. 364] and every locally convex Hausdorff
+space with metric ccp is locally complete by [175, Remark 4.1, p. 267]. All these
+implications are strict. The second by [181, 9-2-10 Example, p. 134] and the others
+by [175, Remark 4.1, p. 267]. For more details on the [metric] convex compactness
+property and local completeness see [29, 175]. In addition, we remark that every
+semi-Montel space is semi-reflexive by [89, 11.5.1 Proposition, p. 230] and every
+semi-reflexive locally convex Hausdorff space is quasi-complete by [153, Chap. IV,
+5.5, Corollary 1, p. 144] and these implications are strict as well. Summarizing, we
+have the following diagram of strict implications:
+semi-Montel ⇒ semi-reflexive
+⇓
+complete ⇒ quasi-complete ⇒ sequentially complete ⇒ locally complete
+⇓
+⇓
+�⇒
+ccp
+⇒
+metric ccp
+Vector-valued continuously partially differentiable functions
+Since weighted spaces of continuously partially differentiable resp. holomorphic
+vector-valued functions will serve as our standard examples, we recall the definition
+of the spaces Ck(Ω,E) resp. O(Ω,E). A function f∶Ω → E on an open set Ω ⊂ Rd
+to an lcHs E is called continuously partially differentiable (f is C1) if for the n-th
+unit vector en ∈ Rd the limit
+(∂en)Ef(x) ∶=
+lim
+h→0
+h∈R,h≠0
+f(x + hen) − f(x)
+h
+exists in E for every x ∈ Ω and (∂en)Ef is continuous on Ω ((∂en)Ef is C0) for
+every 1 ≤ n ≤ d. For k ∈ N a function f is said to be k-times continuously partially
+differentiable (f is Ck) if f is C1 and all its first partial derivatives are Ck−1. A
+function f is called infinitely continuously partially differentiable (f is C∞) if f is
+Ck for every k ∈ N. For k ∈ N∞ ∶= N∪{∞} the functions f∶Ω → E which are Ck form
+a linear space which is denoted by Ck(Ω,E). For β ∈ Nd
+0 with ∣β∣ ∶= ∑d
+n=1 βn ≤ k
+and a function f∶Ω → E on an open set Ω ⊂ Rd to an lcHs E we set (∂βn)Ef ∶= f
+if βn = 0, and
+(∂βn)Ef(x) ∶= (∂en)E⋯(∂en)E
+������������������������������������������������������������������������������������������
+βn-times
+f(x)
+if βn ≠ 0 and the right-hand side exists in E for every x ∈ Ω. Further, we define
+(∂β)Ef(x) ∶= ((∂β1)E⋯(∂βd)E)f(x)
+
+20
+2. NOTATION AND PRELIMINARIES
+if the right-hand side exists in E for every x ∈ Ω. If E = K, we often just write
+∂βf ∶= (∂β)Kf for β ∈ Nd
+0, ∣β∣ ≤ k, and f ∈ Ck(Ω). Furthermore, we define the space
+of bounded continuously partially differentiable functions by
+C1
+b (Ω,E) ∶= {f ∈ C1(Ω,E) ∣ ∀ α ∈ A ∶ ∣f∣C1
+b (Ω),α ∶=
+sup
+x∈Ω
+β∈Nd
+0,∣β∣≤1
+pα((∂β)Ef(x)) < ∞}.
+Vector-valued holomorphic functions
+A function f∶Ω → E on an open set Ω ⊂ C to an lcHs E over C is called
+holomorphic if the limit
+(∂1
+C)Ef(z) ∶=
+lim
+h→0
+h∈C,h≠0
+f(z + h) − f(z)
+h
+,
+z ∈ Ω,
+exists in E.
+We denote by O(Ω,E) the linear space of holomorphic functions
+f∶Ω → E. Defining the vector-valued version of the Cauchy–Riemann operator by
+∂
+Ef ∶= 1
+2((∂e1)E + i(∂e2)E)f
+for f ∈ C(Ω,E) such that the partial derivatives (∂en)Ef, n = 1,2, exist in E, we
+remark that
+O(Ω,E) = {f ∈ C(Ω,E) ∣ f ∈ ker∂
+E} = {f ∈ C∞(Ω,E) ∣ f ∈ ker∂
+E}
+(4)
+by [113, Theorem 6.1, p. 267] if E is locally complete. Further, we set (∂0
+C)Ef ∶= f
+and note that the (n + 1)-th complex derivative (∂n+1
+C
+)Ef ∶= (∂1
+C)E((∂n
+C)Ef) exists
+for all n ∈ N0 and f ∈ O(Ω,E) by [79, 2.1 Theorem and Definition, p. 17–18] and
+[79, 5.2 Theorem, p. 35] if E is locally complete. If E = C, we often just write
+f (n) ∶= (∂n
+C)Cf for n ∈ N0 and f ∈ O(Ω) ∶= O(Ω,C). We note that the real and
+complex derivatives are related by
+(∂β)Ef(z) = iβ2(∂∣β∣
+C )Ef(z),
+z ∈ Ω,
+(5)
+for every f ∈ O(Ω,E) and β = (β1,β2) ∈ N2
+0 by [113, Proposition 7.1, p. 270] if E is
+locally complete.
+
+CHAPTER 3
+The ε-product for weighted function spaces
+3.1. ε-into-compatibility
+In the introduction we already mentioned that linearisations of spaces of vector-
+valued functions by means of ε-products are essential for our approach.
+Here,
+one of the important questions is which spaces of vector-valued functions can be
+represented by ε-products. Let Ω be a non-empty set and E an lcHs. If F(Ω) ⊂ KΩ
+is an lcHs such that δx ∈ F(Ω)′ for all x ∈ Ω, then the map
+S∶F(Ω)εE → EΩ, u �→ [x ↦ u(δx)],
+is well-defined and linear. This leads to the following definition.
+3.1.1. Definition (ε-into-compatible). Let Ω be a non-empty set and E an
+lcHs. Let F(Ω) ⊂ KΩ and F(Ω,E) ⊂ EΩ be lcHs such that δx ∈ F(Ω)′ for all x ∈ Ω.
+We call the spaces F(Ω) and F(Ω,E) ε-into-compatible if the map
+S∶F(Ω)εE → F(Ω,E), u �→ [x ↦ u(δx)],
+is a well-defined isomorphism into, i.e. to its range. We call F(Ω) and F(Ω,E)
+ε-compatible if S is an isomorphism. We write SF(Ω) if we want to emphasise the
+dependency on F(Ω).
+In this section we introduce the weighted space FV(Ω,E) of E-valued functions
+on Ω as a subspace of sections of domains in EΩ of linear operators T E
+m equipped
+with a generalised version of a weighted graph topology. This space is the role
+model for many function spaces and an example for these operators are the partial
+derivative operators. Then we treat the question whether FV(Ω,E) and FV(Ω)εE
+are ε-into-compatible. This is deeply connected with the interplay of the pair of
+operators (T E
+m,T K
+m) with the map S (see Definition 3.1.7). In our main theorem of
+this section we give sufficient conditions such that S∶FV(Ω)εE → FV(Ω,E) is an
+isomorphism into (see Theorem 3.1.12). In the next section we provide conditions
+such that S becomes surjective (see Theorem 3.2.4). We start with the well-known
+example Ck(Ω,E) of k-times continuously partially differentiable E-valued func-
+tions to motivate our definition of FV(Ω,E).
+3.1.2. Example. Let k ∈ N∞ and Ω ⊂ Rd be open. Consider the space C(Ω,E)
+of continuous functions f∶Ω → E with the topology τc of compact convergence, i.e.
+the topology given by the seminorms
+∥f∥K,α ∶= sup
+x∈K
+pα(f(x)),
+f ∈ C(Ω,E),
+for compact K ⊂ Ω and α ∈ A. The usual topology on the space Ck(Ω,E) of k-times
+continuously partially differentiable functions is the graph topology generated by
+the partial derivative operators (∂β)E∶Ck(Ω,E) → C(Ω,E) for β ∈ Nd
+0, ∣β∣ ≤ k, i.e.
+the topology given by the seminorms
+∥f∥K,β,α ∶= max(∥f∥K,α,∥(∂β)Ef∥K,α),
+f ∈ Ck(Ω,E),
+21
+
+22
+3. THE ε-PRODUCT FOR WEIGHTED FUNCTION SPACES
+for compact K ⊂ Ω, β ∈ Nd
+0, ∣β∣ ≤ k, and α ∈ A. The same topology is induced by
+the directed system of seminorms given by
+∣f∣K,m,α ∶=
+sup
+β∈Nd
+0,∣β∣≤m
+∥f∥K,β,α =
+sup
+x∈K
+β∈Nd
+0,∣β∣≤m
+pα((∂β)Ef(x)),
+f ∈ Ck(Ω,E),
+for compact K ⊂ Ω, m ∈ N0, m ≤ k, and α ∈ A and may also be seen as a weighted
+topology induced by the family (χK) of characteristic functions of the compact sets
+K ⊂ Ω by writing
+∣f∣K,m,α =
+sup
+x∈Ω
+β∈Nd
+0,∣β∣≤m
+pα((∂β)Ef(x))χK(x),
+f ∈ Ck(Ω,E).
+This topology is inherited by linear subspaces of functions having additional prop-
+erties like being holomorphic or harmonic.
+We turn to the weight functions which we use to define a kind of weighted
+graph topology.
+3.1.3. Definition (weight function). Let J be a non-empty set and (ωm)m∈M
+a family of non-empty sets. We call V ∶= (νj,m)j∈J,m∈M a family of weight functions
+on (ωm)m∈M if it fulfils νj,m∶ωm → [0,∞) for all j ∈ J, m ∈ M and
+∀ m ∈ M, x ∈ ωm ∃ j ∈ J ∶ 0 < νj,m(x).
+(6)
+From the structure of Example 3.1.2 we arrive at the following definition of the
+weighted spaces of vector-valued functions we want to consider.
+3.1.4. Definition. Let Ω be a non-empty set, V ∶= (νj,m)j∈J,m∈M a family
+of weight functions on (ωm)m∈M and T E
+m∶EΩ ⊃ domT E
+m → Eωm a linear map for
+every m ∈ M. Let AP(Ω,E) be a linear subspace of EΩ and define the space of
+intersections
+F(Ω,E) ∶= AP(Ω,E) ∩ ( ⋂
+m∈M
+domT E
+m)
+as well as
+FV(Ω,E) ∶= {f ∈ F(Ω,E) ∣ ∀ j ∈ J, m ∈ M, α ∈ A ∶ ∣f∣j,m,α < ∞}
+where
+∣f∣j,m,α ∶= sup
+x∈ωm
+pα(T E
+m(f)(x))νj,m(x) =
+sup
+e∈Nj,m(f)
+pα(e)
+with
+Nj,m(f) ∶= {T E
+m(f)(x)νj,m(x) ∣ x ∈ ωm}.
+Further, we write F(Ω) ∶= F(Ω,K) and FV(Ω) ∶= FV(Ω,K). If we want to empha-
+sise dependencies, we write M(E) instead of M, APFV(Ω,E) instead of AP(Ω,E)
+and ∣f∣FV(Ω),j,m,α instead of ∣f∣j,m,α. If J, M or A are singletons, we omit the index
+j, m resp. α in ∣f∣j,m,α.
+Note that ωm need not be a subset of Ω. The space AP(Ω,E) is a placeholder
+where we collect additional properties (AP) of our functions not being reflected by
+the operators T E
+m which we integrated in the topology. However, these additional
+properties might come from being in the domain or kernel of additional operators,
+e.g. harmonicity means being in the kernel of the Laplacian. But often AP(Ω,E)
+can be chosen as EΩ or C(Ω,E). The space FV(Ω,E) is locally convex but need not
+be Hausdorff. Since it is easier to work with Hausdorff spaces and a directed family
+of seminorms plus the point evaluation functionals δx∶FV(Ω) → K, f ↦ f(x), for
+x ∈ Ω and their continuity play a big role, we introduce the following definition.
+
+3.1. ε-INTO-COMPATIBILITY
+23
+3.1.5. Definition (dom-space and T E
+m,x). We call FV(Ω,E) a dom-space if it
+is a locally convex Hausdorff space, the system of seminorms (∣f∣j,m,α)j∈J,m∈M,α∈A
+is directed and, in addition, δx ∈ FV(Ω)′ for every x ∈ Ω if E = K. We define the
+point evaluation of T E
+m by T E
+m,x∶domT E
+m → E, T E
+m,x(f) ∶= T E
+m(f)(x), for m ∈ M and
+x ∈ ωm.
+3.1.6. Remark.
+a) It is easy to see that FV(Ω,E) is Hausdorff if there
+is m ∈ M such that ωm = Ω and T E
+m = idEΩ since E is Hausdorff.
+b) If E = K, then T K
+m,x ∈ FV(Ω)′ for every m ∈ M and x ∈ ωm. Indeed, for
+m ∈ M and x ∈ ωm there exists j ∈ J such that νj,m(x) > 0 by (6), implying
+for every f ∈ FV(Ω) that
+∣T K
+m,x(f)∣ =
+1
+νj,m(x)∣T K
+m(f)(x)∣νj,m(x) ≤
+1
+νj,m(x)∣f∣j,m.
+In particular, this implies δx ∈ FV(Ω)′ for all x ∈ Ω if there is m ∈ M such
+that ωm = Ω and T K
+m = idKΩ.
+c) Let the family of weight functions V be directed, i.e.
+∀ j1,j2 ∈ J,m1,m2 ∈ M ∃ j3 ∈ J, m3 ∈ M, C > 0 ∀ i ∈ {1,2} ∶
+(ωm1 ∪ ωm2) ⊂ ωm3
+and
+νji,mi ≤ Cνj3,m3.
+Then the system of seminorms (∣f∣j,m,α)j∈J,m∈M,α∈A is directed if V is
+directed and additionally it holds with mi, i ∈ {1,2,3}, from above that
+∀ f ∈ FV(Ω,E), i ∈ {1,2}, x ∈ ωmi ∶ T E
+mi(f)(x) = T E
+m3(f)(x),
+since the system (pα)α∈A of E is already directed.
+We point out that the additional condition in Remark 3.1.6 c) is missing in
+[110, Remark 5 c), p. 1516] (resp. [106, 3.5 Remark, p. 6]), which we correct here.
+For the lcHs E over K we want to define a natural E-valued version of a dom-
+space FV(Ω) = FV(Ω,K). The natural E-valued version of FV(Ω) should be a
+dom-space FV(Ω,E) such that there is a canonical relation between the families
+(T K
+m) and (T E
+m). This canonical relation will be explained in terms of their interplay
+with the map
+S∶FV(Ω)εE → EΩ, u �→ [x ↦ u(δx)].
+Further, the elements of our E-valued version FV(Ω,E) of FV(Ω) should be com-
+patible with a weak definition in the sense that e′ ○f ∈ FV(Ω) should hold for every
+e′ ∈ E′ and f ∈ FV(Ω,E).
+3.1.7. Definition (generator, consistent, strong). Let FV(Ω) and FV(Ω,E)
+be dom-spaces such that M ∶= M(K) = M(E).
+a) We call (T E
+m,T K
+m)m∈M a generator for (FV(Ω),E), in short, (FV,E).
+b) We call (T E
+m,T K
+m)m∈M consistent if we have for all u ∈ FV(Ω)εE that
+S(u) ∈ F(Ω,E) and
+∀ m ∈ M, x ∈ ωm ∶ (T E
+mS(u))(x) = u(T K
+m,x).
+c) We call (T E
+m,T K
+m)m∈M strong if we have for all e′ ∈ E′, f ∈ FV(Ω,E) that
+e′ ○ f ∈ F(Ω) and
+∀ m ∈ M, x ∈ ωm ∶ T K
+m(e′ ○ f)(x) = (e′ ○ T E
+m(f))(x).
+More precisely, T K
+m,x in b) means the restriction of T K
+m,x to FV(Ω) and the
+term u(T K
+m,x) is well-defined by Remark 3.1.6 b). Consistency will guarantee that
+the map S∶FV(Ω)εE → FV(Ω,E) is a well-defined isomorphism into, i.e. ε-into-
+compatibility, and strength will help us to prove its surjectivity under some ad-
+ditional assumptions on FV(Ω) and E. Let us come to a lemma which describes
+
+24
+3. THE ε-PRODUCT FOR WEIGHTED FUNCTION SPACES
+the topology of FV(Ω)εE in terms of the operators T K
+m with m ∈ M. It was the
+motivation for the definition of consistency and allows us to consider FV(Ω)εE as
+a topological subspace of FV(Ω,E) via S, assuming consistency.
+3.1.8. Lemma. Let FV(Ω) be a dom-space. Then the topology of FV(Ω)εE is
+given by the system of seminorms defined by
+∥u∥j,m,α ∶= sup
+x∈ωm
+pα(u(T K
+m,x))νj,m(x),
+u ∈ FV(Ω)εE,
+for j ∈ J, m ∈ M and α ∈ A.
+Proof. We define the sets Dj,m ∶= {T K
+m,x(⋅)νj,m(x) ∣ x ∈ ωm} and Bj,m ∶= {f ∈
+FV(Ω) ∣ ∣f∣j,m ≤ 1} for every j ∈ J and m ∈ M. We claim that acx(Dj,m) is dense
+in the polar B○
+j,m with respect to κ(FV(Ω)′,FV(Ω)). The observation
+D○
+j,m = {T K
+m,x(⋅)νj,m(x) ∣ x ∈ ωm}○
+= {f ∈ FV(Ω) ∣ ∀x ∈ ωm ∶ ∣T K
+m(f)(x)∣νj,m(x) ≤ 1}
+= {f ∈ FV(Ω) ∣ ∣f∣j,m ≤ 1} = Bj,m
+yields
+acx(Dj,m)κ(FV(Ω)′,FV(Ω)) = (Dj,m)○○ = B○
+j,m
+by the bipolar theorem. By [89, 8.4, p. 152, 8.5, p. 156–157] the system of seminorms
+defined by
+qj,m,α(u) ∶= sup
+y∈B○
+j,m
+pα(u(y)),
+u ∈ FV(Ω)εE,
+for j ∈ J, m ∈ M and α ∈ A gives the topology on FV(Ω)εE (here it is used that
+the system of seminorms (∣ ⋅ ∣j,m) of FV(Ω) is directed). As every u ∈ FV(Ω)εE
+is continuous on B○
+j,m, we may replace B○
+j,m by a κ(FV(Ω)′,FV(Ω))-dense subset.
+Therefore we obtain
+qj,m,α(u) = sup{pα(u(y)) ∣ y ∈ acx(Dj,m)}.
+For y ∈ acx(Dj,m) there are n ∈ N, λk ∈ K, xk ∈ ωm, 1 ≤ k ≤ n, with ∑n
+k=1 ∣λk∣ ≤ 1
+such that y = ∑n
+k=1 λkT K
+m,xk(⋅)νj,m(xk). Then we have for every u ∈ FV(Ω)εE
+pα(u(y)) ≤
+n
+∑
+k=1
+∣λk∣pα(u(T K
+m,xk))νj,m(xk) ≤ ∥u∥j,m,α,
+thus qj,m,α(u) ≤ ∥u∥j,m,α. On the other hand, we derive
+qj,m,α(u) ≥ sup
+y∈Dj,m
+pα(u(y)) = sup
+x∈ωm
+pα(u(T K
+m,x))νj,m(x) = ∥u∥j,m,α.
+□
+Let us turn to a more general version of Example 3.1.2, namely, to weighted
+spaces of k-times continuously partially differentiable functions and kernels of linear
+partial differential operators in these spaces.
+3.1.9. Example. Let k ∈ N∞ and Ω ⊂ Rd be open. We consider the cases
+(i) ωm ∶= Mm × Ω with Mm ∶= {β ∈ Nd
+0 ∣ ∣β∣ ≤ min(m,k)} for all m ∈ N0, or
+(ii) ωm ∶= Nd
+0 × Ω for all m ∈ N0 and k = ∞,
+and let Vk ∶= (νj,m)j∈J,m∈N0 be a directed family of weights on (ωm)m∈N0.
+a) We define the weighted space of k-times continuously partially differentiable
+functions with values in an lcHs E as
+CVk(Ω,E) ∶= {f ∈ Ck(Ω,E) ∣ ∀ j ∈ J, m ∈ N0, α ∈ A ∶ ∣f∣j,m,α < ∞}
+where
+∣f∣j,m,α ∶=
+sup
+(β,x)∈ωm
+pα((∂β)Ef(x))νj,m(β,x).
+
+3.1. ε-INTO-COMPATIBILITY
+25
+Setting domT E
+m ∶= Ck(Ω,E) and
+T E
+m∶Ck(Ω,E) → Eωm, f �→ [(β,x) ↦ (∂β)Ef(x)],
+as well as AP(Ω,E) ∶= EΩ, we observe that CVk(Ω,E) is a dom-space by Remark
+3.1.6 and
+∣f∣j,m,α = sup
+x∈ωm
+pα(T E
+mf(x))νj,m(x).
+b) The space Ck(Ω,E) with its usual topology given in Example 3.1.2 is a special
+case of a)(i) with J ∶= {K ⊂ Ω ∣ K compact}, νK,m(β,x) ∶= χK(x), (β,x) ∈ ωm, for
+all m ∈ N0 and K ∈ J where χK is the characteristic function of K. In this case we
+write Wk ∶= Vk for the family of weight functions.
+c) The Schwartz space is defined by
+S(Rd,E) ∶= {f ∈ C∞(Rd,E) ∣ ∀ m ∈ N0, α ∈ A ∶ ∣f∣m,α < ∞}
+where
+∣f∣m,α ∶=
+sup
+x∈Rd
+β∈Nd
+0,∣β∣≤m
+pα((∂β)Ef(x))(1 + ∣x∣2)m/2.
+This is a special case of a)(i) with k ∶= ∞, Ω ∶= Rd, J ∶= {1} and ν1,m(β,x) ∶=
+(1 + ∣x∣2)m/2, (β,x) ∈ ωm, for all m ∈ N0.
+d) The multiplier space for the Schwartz space is defined by
+OM(Rd,E) ∶= {f ∈ C∞(Rd,E) ∣ ∀ g ∈ S(Rd), m ∈ N0, α ∈ A ∶ ∥f∥g,m,α < ∞}
+where
+∥f∥g,m,α ∶=
+sup
+x∈Rd
+β∈Nd
+0,∣β∣≤m
+pα((∂β)Ef(x))∣g(x)∣
+(see [158, 40), p. 97]). This is a special case of a)(i) with k ∶= ∞, Ω ∶= Rd, J ∶=
+{j ⊂ S(Rd) ∣ j finite} and νj,1,m(β,x) ∶= maxg∈j ∣g(x)∣, (β,x) ∈ ωm, for all m ∈ N0.
+This choice of J guarantees that the family V∞ is directed and does not change the
+topology.
+e) Let K ∶= {K ⊂ Ω ∣ K compact} and (Mp)p∈N0 be a sequence of positive real
+numbers. The space E(Mp)(Ω,E) of ultradifferentiable functions of class (Mp) of
+Beurling-type is defined as
+E(Mp)(Ω,E) ∶= {f ∈ C∞(Ω,E) ∣ ∀ K ∈ K, h > 0, α ∈ A ∶ ∣f∣(K,h),α < ∞}
+where
+∣f∣(K,h),α ∶= sup
+x∈K
+β∈Nd
+0
+pα((∂β)Ef(x))
+1
+h∣β∣M∣β∣
+.
+This is a special case of a)(ii) with J ∶= K × R>0 and ν(K,h),m(β,x) ∶= χK(x)
+1
+h∣β∣M∣β∣ ,
+(β,x) ∈ ωm, for all (K,h) ∈ J and m ∈ N0 where R>0 ∶= (0,∞).
+f) Let K and (Mp)p∈N0 be as in e). The space E{Mp}(Ω,E) of ultradifferentiable
+functions of class {Mp} of Roumieu-type is defined as
+E{Mp}(Ω,E) ∶= {f ∈ C∞(Ω,E) ∣ ∀ (K,H) ∈ J, α ∈ A ∶ ∣f∣(K,H),α < ∞}
+where
+J ∶= K × {H = (Hn)n∈N ∣ ∃ (hk)k∈N, hk > 0, hk ↗ ∞ ∀ n ∈ N ∶ Hn = h1 ⋅ ... ⋅ hn}
+and
+∣f∣(K,H),α ∶= sup
+x∈K
+β∈Nd
+0
+pα((∂β)Ef(x))
+1
+H∣β∣M∣β∣
+
+26
+3. THE ε-PRODUCT FOR WEIGHTED FUNCTION SPACES
+(see [101, Proposition 3.5, p. 675]).
+Again, this is a special case of a)(ii) with
+ν(K,H),m(β,x) ∶= χK(x)
+1
+H∣β∣M∣β∣ , (β,x) ∈ ωm, for all (K,H) ∈ J and m ∈ N0.
+g) Let n ∈ N, βi ∈ Nd
+0 with ∣βi∣ ≤ k and ai∶Ω → K for 1 ≤ i ≤ n. We set
+P(∂)E∶Ck(Ω,E) → EΩ, P(∂)E(f)(x) ∶=
+n
+∑
+i=1
+ai(x)(∂βi)E(f)(x)
+and obtain the (topological) subspace of CVk(Ω,E) given by
+CVk
+P (∂)(Ω,E) ∶= {f ∈ CVk(Ω,E) ∣ f ∈ kerP(∂)E}.
+Choosing AP(Ω,E) ∶= kerP(∂)E, we see that this is also a dom-space by a). If
+P(∂)E is the Cauchy–Riemann operator (and E locally complete) or the Laplacian,
+we obtain the weighted space of holomorphic resp. harmonic functions.
+Let us show that the generators of these spaces are strong and consistent. In
+order to obtain consistency for their generators we have to restrict to directed
+families of weights which are locally bounded away from zero on Ω, i.e.
+∀ K ⊂ Ω compact, m ∈ N0 ∃ j ∈ J ∀ β ∈ Nd
+0, ∣β∣ ≤ min(m,k) ∶ inf
+x∈K νj,m(β,x) > 0.
+This condition on Vk guarantees that the map I∶CVk(Ω) → CWk(Ω), f ↦ f, is
+continuous which is needed for consistency.
+3.1.10. Proposition. Let E be an lcHs, k ∈ N∞, Vk be a directed family of
+weights which is locally bounded away from zero on an open set Ω ⊂ Rd.
+The
+generator of (CVk,E) resp. (CVk
+P (∂),E) from Example 3.1.9 is strong and consistent
+if CVk(Ω) resp. CVk
+P (∂)(Ω) is barrelled.
+Proof. We recall the definitions from Example 3.1.9. We have ωm ∶= Mm × Ω
+with Mm ∶= {β ∈ Nd
+0 ∣ ∣β∣ ≤ min(m,k)} for all m ∈ N0 or ωm ∶= Nd
+0 × Ω for all m ∈ N0.
+Further, APCVk(Ω,E) = EΩ, APCVk
+P (∂)(Ω,E) = kerP(∂)E, domT E
+m ∶= Ck(Ω,E) and
+T E
+m∶Ck(Ω,E) → Eωm, f �→ [(β,x) ↦ (∂β)Ef(x)],
+for all m ∈ N0 and the same with K instead of E. The family (T E
+m,T K
+m)m∈N0 is a
+strong generator for (CVk,E) because
+(∂β)K(e′ ○ f)(x) = e′((∂β)Ef(x)),
+(β,x) ∈ ωm,
+for all e′ ∈ E′, f ∈ CVk(Ω,E) and m ∈ N0 due to the linearity and continuity of
+e′ ∈ E′. In addition, e′ ○ f ∈ kerP(∂)K for all e′ ∈ E′ and f ∈ CVk
+P (∂)(Ω,E), which
+implies that (T E
+m,T K
+m)m∈N0 is also a strong generator for (CVk
+P (∂),E).
+For consistency we need to prove that
+(∂β)ES(u)(x) = u(δx ○ (∂β)K),
+(β,x) ∈ ωm,
+for all u ∈ CVk(Ω)εE resp. u ∈ CVk
+P (∂)(Ω)εE. This follows from the subsequent
+Proposition 3.1.11 b) since FV(Ω) = CVk(Ω) resp. FV(Ω) = CVk
+P (∂)(Ω) is barrelled
+and Vk locally bounded away from zero on Ω. Thus (T E
+m,T K
+m)m∈N0 is a consistent
+generator for (CVk,E). In addition, we have with P(∂)E from Example 3.1.9 g)
+that
+P(∂)E(S(u))(x) =
+n
+∑
+i=1
+ai(x)(∂βi)E(S(u))(x) = u(
+n
+∑
+i=1
+ai(x)(δx ○ (∂βi)K))
+= u(δx ○ P(∂)K) = 0,
+x ∈ Ω,
+(7)
+for every u ∈ CVk
+P (∂)(Ω)εE. This yields S(u) ∈ kerP(∂)E for all u ∈ CVk
+P (∂)(Ω)εE.
+Therefore (T E
+m,T K
+m)m∈N0 is a consistent generator for (CVk
+P (∂),E) as well.
+□
+
+3.1. ε-INTO-COMPATIBILITY
+27
+Let us turn to the postponed part in the proof of consistency. We denote by
+CW(Ω) the space of scalar-valued continuous functions on an open set Ω ⊂ Rd
+with the topology of uniform convergence on compact subsets, i.e. the weighted
+topology given by the family of weights W ∶= W0 ∶= {χK ∣ K ⊂ Ω compact}, and we
+set δ(x) ∶= δx for x ∈ Ω.
+3.1.11. Proposition. Let Ω ⊂ Rd be open, k ∈ N∞ and FV(Ω) a dom-space.
+a) If T ∈ L(FV(Ω),CW(Ω)), then δ ○ T ∈ C(Ω,FV(Ω)′
+γ).
+b) If T ∈ L(FV(Ω),CW1(Ω)) and FV(Ω) is barrelled, then
+(∂en)FV(Ω)′
+κ(δ ○ T)(x) = lim
+h→0
+δx+hen ○ T − δx ○ T
+h
+= δx ○ (∂en)K ○ T,
+x ∈ Ω, 1 ≤ n ≤ d,
+and δ ○ T ∈ C1(Ω,FV(Ω)′
+κ).
+c) If the inclusion I∶FV(Ω) → CWk(Ω), f ↦ f, is continuous and FV(Ω)
+barrelled, then S(u) ∈ Ck(Ω,E) and
+(∂β)ES(u)(x) = u(δx ○ (∂β)K),
+β ∈ Nd
+0, ∣β∣ ≤ k, x ∈ Ω,
+for all u ∈ FV(Ω)εE.
+Proof. a) First, if x ∈ Ω and (xτ)τ∈T is a net in Ω converging to x, then we
+observe that
+(δxτ ○ T)(f) = T(f)(xτ) → T(f)(x) = (δx ○ T)(f)
+for every f ∈ FV(Ω) as T(f) is continuous on Ω. Second, let K ⊂ Ω be compact.
+Then there are j ∈ J, m ∈ M and C > 0 such that
+sup
+x∈K
+∣(δx ○ T)(f)∣ = sup
+x∈K
+∣T(f)(x)∣ ≤ C∣f∣j,m
+for every f ∈ FV(Ω).
+This means that {δx ○ T ∣ x ∈ K} is equicontinuous in
+FV(Ω)′. The topologies σ(FV(Ω)′,FV(Ω)) and γ(FV(Ω)′,FV(Ω)) coincide on
+equicontinuous subsets of FV(Ω)′, implying that the restriction (δ ○ T)∣K∶K →
+FV(Ω)′
+γ is continuous by our first observation. As δ ○ T is continuous on every
+compact subset of the open set Ω ⊂ Rd, it follows that δ ○ T∶Ω → FV(Ω)′
+γ is well-
+defined and continuous.
+b) Let x ∈ Ω and 1 ≤ n ≤ d. Then there is ε > 0 such that x + hen ∈ Ω for all
+h ∈ R with 0 < ∣h∣ < ε. We note that δ ○ T ∈ C(Ω,FV(Ω)′
+κ) by part a), which implies
+δx+hen○T −δx○T
+h
+∈ FV(Ω)′. For every f ∈ FV(Ω) we have
+lim
+h→0
+δx+hen ○ T − δx ○ T
+h
+(f) = lim
+h→0
+T(f)(x + hen) − T(f)(x)
+h
+= (∂en)KT(f)(x)
+in K as T(f) ∈ C1(Ω). Therefore 1
+h(δx+hen ○ T − δx ○ T) converges to δx ○ (∂en)K ○ T
+in FV(Ω)′
+σ and thus in FV(Ω)′
+κ by the Banach–Steinhaus theorem as well.
+In
+particular, we obtain
+δx ○ (∂en)K ○ T = lim
+h→0
+δx+hen ○ T − δx ○ T
+h
+= (∂en)FV(Ω)′
+κ(δ ○ T)(x)
+in FV(Ω)′
+κ. Moreover, δ ○ (∂en)K ○ T ∈ C(Ω,FV(Ω)′
+κ) by part a) as (∂en)K ○ T ∈
+L(FV(Ω),CW(Ω)). Hence we deduce that δ ○ T ∈ C1(Ω,FV(Ω)′
+κ).
+c) We prove our claim by induction on the order of differentiation. Let u ∈
+FV(Ω)εE. For β ∈ Nd
+0 with ∣β∣ = 0 we get S(u) = u ○ δ ∈ C(Ω,E) from part a) with
+T = I. Further,
+(∂β)ES(u)(x) = S(u)(x) = u(δx) = u(δx ○ (∂β)K),
+x ∈ Ω.
+
+28
+3. THE ε-PRODUCT FOR WEIGHTED FUNCTION SPACES
+Let m ∈ N0, m < k, such that S(u) ∈ Cm(Ω,E) and
+(∂β)ES(u)(x) = u(δx ○ (∂β)K),
+x ∈ Ω,
+(8)
+for all β ∈ Nd
+0 with ∣β∣ ≤ m.
+Let β ∈ Nd
+0 with ∣β∣ = m + 1 ≤ k.
+Then there is
+1 ≤ n ≤ d and ̃β ∈ Nd
+0 with ∣̃β∣ = m such that β = en + ̃β. The barrelledness of
+FV(Ω) yields that 1
+h(δx+hen ○ (∂
+̃β)K − δx ○ (∂
+̃β)K) converges to δx ○ (∂en)K ○ (∂
+̃β)K
+in FV(Ω)′
+κ for every x ∈ Ω by part b) with T ∶= (∂
+̃β)K. Therefore we derive from
+δx ○ (∂en)K ○ (∂
+̃β)K = δx ○ (∂β)K by Schwarz’ theorem that
+u(δx ○ (∂β)K) = lim
+h→0
+1
+h(u(δx+hen ○ (∂
+̃β)K) − u(δx ○ (∂
+̃β)K))
+=
+(8) lim
+h→0
+1
+h((∂
+̃β)ES(u)(x + hen) − (∂
+̃β)ES(u)(x))
+= (∂en)E(∂
+̃β)ES(u)(x)
+for every x ∈ Ω. Moreover, δ ○ (∂β)K = (∂en)FV(Ω)′
+κ(δ ○ T) ∈ C(Ω,FV(Ω)′
+κ) for
+T = (∂
+̃β)K by part b).
+Hence we have S(u) ∈ Cm+1(Ω,E) and it follows from
+Schwarz’ theorem again that
+u(δx ○ (∂β)K) = (∂en)E(∂
+̃β)ES(u)(x) = (∂β)ES(u)(x),
+x ∈ Ω.
+□
+Part a) of the preceding proposition is just a modification of [16, 4.1 Lemma, p.
+198], where FV(Ω) = CV(Ω) is the Nachbin-weighted space of continuous functions
+and T = id, and holds more general for kR-spaces Ω (see Lemma 4.1.2).
+3.1.12. Theorem. Let (T E
+m,T K
+m)m∈M be a consistent generator for (FV,E).
+Then the map S∶FV(Ω)εE → FV(Ω,E) is an isomorphism into, i.e. the spaces
+FV(Ω) and FV(Ω,E) are ε-into-compatible.
+Proof. First, we show that S(FV(Ω)εE) ⊂ FV(Ω,E). Let u ∈ FV(Ω)εE.
+Due to the consistency of (T E
+m,T K
+m)m∈M we have S(u) ∈ AP(Ω,E) ∩ domT E
+m and
+(T E
+mS(u))(x) = u(T K
+m,x),
+m ∈ M x ∈ ωm.
+Furthermore, we get by Lemma 3.1.8 for every j ∈ J, m ∈ M and α ∈ A
+∣S(u)∣j,m,α = sup
+x∈ωm
+pα(T E
+m(S(u))(x))νj,m(x) = ∥u∥j,m,α < ∞,
+(9)
+implying S(u) ∈ FV(Ω,E) and the continuity of S. Moreover, we deduce from (9)
+that S is injective and that the inverse of S on the range of S is also continuous.
+□
+3.1.13. Remark. If J, M and A are countable, then S is an isometry with
+respect to the induced metrics on FV(Ω,E) and FV(Ω)εE by (9).
+The basic idea for Theorem 3.1.12 was derived from analysing the proof of an
+analogous statement for Bierstedt’s weighted spaces CV(Ω,E) and CV0(Ω,E) of
+continuous functions already mentioned in the introduction (see [16, 4.2 Lemma,
+4.3 Folgerung, p. 199–200] and [17, 2.1 Satz, p. 137]).
+3.2. ε-compatibility
+Now, we try to answer the natural question. When is S surjective? The strength
+of a generator and a weaker concept to define a natural E-valued version of FV(Ω)
+come into play to answer the question on the surjectivity of our key map S. Let
+FV(Ω) be a dom-space. We define the linear space of E-valued weak FV-functions
+by
+FV(Ω,E)σ ∶= {f∶Ω → E ∣ ∀ e′ ∈ E′ ∶ e′ ○ f ∈ FV(Ω)}.
+
+3.2. ε-COMPATIBILITY
+29
+Moreover, for f ∈ FV(Ω,E)σ we define the linear map
+Rf∶E′ → FV(Ω), Rf(e′) ∶= e′ ○ f,
+and the dual map
+Rt
+f∶FV(Ω)′ → E′⋆, f ′ �→ [e′ ↦ f ′(Rf(e′))],
+where E′⋆ is the algebraic dual of E′. Furthermore, we set
+FV(Ω,E)κ ∶= {f ∈ FV(Ω,E)σ ∣ ∀ α ∈ A ∶ Rf(B○
+α) relatively compact in FV(Ω)}
+where Bα ∶= {x ∈ E ∣ pα(x) < 1} for α ∈ A. Next, we give a sufficient condition for
+the inclusion FV(Ω,E) ⊂ FV(Ω,E)σ by means of the family (T E
+m,T K
+m)m∈M.
+3.2.1. Lemma. If (T E
+m,T K
+m)m∈M is a strong generator for (FV,E), then we have
+FV(Ω,E) ⊂ FV(Ω,E)σ and
+sup
+e′∈B○α
+∣Rf(e′)∣j,m = ∣f∣j,m,α
+(10)
+for every f ∈ FV(Ω,E), j ∈ J, m ∈ M and α ∈ A.
+Proof. Let f ∈ FV(Ω,E).
+We have e′ ○ f ∈ F(Ω) for every e′ ∈ E′ since
+(T E
+m,T K
+m)m∈M is a strong generator. Moreover, we have
+∣Rf(e′)∣j,m = ∣e′ ○ f∣j,m = sup
+x∈ωm
+∣T K
+m(e′ ○ f)(x)∣νj,m(x)
+= sup
+x∈ωm
+∣e′(T E
+m(f)(x))∣νj,m(x) =
+sup
+x∈Nj,m(f)
+∣e′(x)∣
+(11)
+for every j ∈ J and m ∈ M with the set Nj,m(f) from Definition 3.1.4. We note
+that Nj,m(f) is bounded in E by Definition 3.1.4 and thus weakly bounded, im-
+plying that the right-hand side of (11) is finite. Hence we conclude f ∈ FV(Ω,E)σ.
+Further, we observe that
+sup
+e′∈B○α
+∣Rf(e′)∣j,m = ∣f∣j,m,α
+for every j ∈ J, m ∈ M and α ∈ A due to [131, Proposition 22.14, p. 256].
+□
+Now, we phrase some sufficient conditions for FV(Ω,E) ⊂ FV(Ω,E)κ to hold
+which is one of the key points regarding the surjectivity of S.
+3.2.2. Lemma. If (T E
+m,T K
+m)m∈M is a strong generator for (FV,E) and one of
+the following conditions is fulfilled, then FV(Ω,E) ⊂ FV(Ω,E)κ.
+a) FV(Ω) is a semi-Montel space.
+b) E is a semi-Montel or Schwartz space.
+c) ∀ f ∈ FV(Ω,E), j ∈ J, m ∈ M ∃ K ∈ γ(E) ∶ Nj,m(f) ⊂ K.
+Proof. Let f ∈ FV(Ω,E). By virtue of Lemma 3.2.1 we already have f ∈
+FV(Ω,E)σ.
+a) For every j ∈ J, m ∈ M and α ∈ A we derive from
+sup
+e′∈B○α
+∣Rf(e′)∣j,m =
+(10) ∣f∣j,m,α < ∞
+that Rf(B○
+α) is bounded and thus relatively compact in the semi-Montel space
+FV(Ω).
+c) It follows from (11) that Rf ∈ L(E′
+γ,FV(Ω)).
+Further, the polar B○
+α is
+relatively compact in E′
+γ for every α ∈ A by the Alaoğlu–Bourbaki theorem. The
+continuity of Rf implies that Rf(B○
+α) is relatively compact as well.
+b) Let j ∈ J and m ∈ M. The set K ∶= Nj,m(f) is bounded in E by Definition
+3.1.4. We deduce that K is already precompact in E by [89, 10.4.3 Corollary, p.
+202] if E is a Schwartz space resp. since it is relatively compact if E is a semi-Montel
+space. Hence the statement follows from c).
+□
+
+30
+3. THE ε-PRODUCT FOR WEIGHTED FUNCTION SPACES
+Let us turn to sufficient conditions for FV(Ω,E) ≅ FV(Ω)εE. For the lcHs E
+we denote by J ∶E → E′⋆, x �→ [e′ ↦ e′(x)], the canonical injection.
+3.2.3. Condition. Let (T E
+m,T K
+m)m∈M be a strong generator for (FV,E). Define
+the following conditions:
+a) E is complete.
+b) E is quasi-complete and for every f ∈ FV(Ω,E) and f ′ ∈ FV(Ω)′ there is
+a bounded net (f ′
+τ)τ∈T in FV(Ω)′ converging to f ′ in FV(Ω)′
+κ such that
+Rt
+f(f ′
+τ) ∈ J (E) for every τ ∈ T .
+c) E is sequentially complete and for every f ∈ FV(Ω,E) and f ′ ∈ FV(Ω)′
+there is a sequence (f ′
+n)n∈N in FV(Ω)′ converging to f ′ in FV(Ω)′
+κ such
+that Rt
+f(f ′
+n) ∈ J (E) for every n ∈ N.
+d) E is locally complete and for every f ∈ FV(Ω,E) and f ′ ∈ FV(Ω)′ there
+is a sequence (f ′
+n)n∈N in FV(Ω)′ locally converging to f ′ in FV(Ω)′
+κ such
+that Rt
+f(f ′
+n) ∈ J (E) for every n ∈ N.
+e) ∀ f ∈ FV(Ω,E), j ∈ J, m ∈ M ∃ K ∈ τ(E) ∶ Nj,m(f) ⊂ K.
+3.2.4. Theorem. Let (T E
+m,T K
+m)m∈M be a consistent generator for (FV,E) and
+let FV(Ω,E) ⊂ FV(Ω,E)κ. If one of the Conditions 3.2.3 is fulfilled, then the
+map S∶FV(Ω)εE → FV(Ω,E) is an isomorphism, i.e. FV(Ω) and FV(Ω,E) are
+ε-compatible. The inverse of S is given by the map
+Rt∶FV(Ω,E) → FV(Ω)εE, f ↦ J −1 ○ Rt
+f,
+where J ∶E → E′⋆ is the canonical injection and
+Rt
+f∶FV(Ω)′ → E′⋆, f ′ �→ [e′ ↦ f ′(Rf(e′))],
+with Rf(e′) = e′ ○ f.
+Proof. Due to Theorem 3.1.12 we only have to show that S is surjective. We
+equip J (E) with the system of seminorms given by
+pB○α(J (x)) ∶= sup
+e′∈B○α
+∣J (x)(e′)∣ = pα(x),
+x ∈ E,
+(12)
+for every α ∈ A. Let f ∈ FV(Ω,E). We consider the dual map Rt
+f and claim that
+Rt
+f ∈ L(FV(Ω)′
+κ,J (E)). Indeed, we have
+pB○α(Rt
+f(y)) = sup
+e′∈B○α
+∣y(Rf(e′))∣ =
+sup
+x∈Rf (B○α)
+∣y(x)∣ ≤ sup
+x∈Kα
+∣y(x)∣
+(13)
+for all y ∈ FV(Ω)′ where Kα ∶= Rf(B○α). Since FV(Ω,E) ⊂ FV(Ω,E)κ, the set
+Rf(B○
+α) is absolutely convex and relatively compact, implying that Kα is absolutely
+convex and compact in FV(Ω) by [89, 6.2.1 Proposition, p. 103]. Further, we have
+for all e′ ∈ E′ and x ∈ Ω
+Rt
+f(δx)(e′) = δx(e′ ○ f) = e′(f(x)) = J (f(x))(e′)
+(14)
+and thus Rt
+f(δx) ∈ J (E).
+a) Let E be complete and f ′ ∈ FV(Ω)′. Since the span of {δx ∣ x ∈ Ω} is dense
+in F(Ω)′
+κ by the bipolar theorem, there is a net (f ′
+τ) converging to f ′ in FV(Ω)′
+κ
+with Rt
+f(f ′
+τ) ∈ J (E) by (14). As
+pB○α(Rt
+f(f ′
+τ) − Rt
+f(f ′)) ≤
+(13) sup
+x∈Kα
+∣(f ′
+τ − f ′)(x)∣ → 0,
+(15)
+for all α ∈ A, we gain that (Rt
+f(f ′
+τ)) is a Cauchy net in the complete space J (E).
+Hence it has a limit g ∈ J (E) which coincides with Rt
+f(f ′) since
+pB○α(g − Rt
+f(f ′)) ≤ pB○α(g − Rt
+f(f ′
+τ)) + pB○α(Rt
+f(f ′
+τ) − Rt
+f(f ′))
+
+3.2. ε-COMPATIBILITY
+31
+≤
+(15)pB○α(g − Rt
+f(f ′
+τ)) + sup
+x∈Kα
+∣(f ′
+τ − f ′)(x)∣ → 0
+for all α ∈ A. We conclude that Rt
+f(f ′) ∈ J (E) for every f ′ ∈ FV(Ω)′.
+b) Let Condition 3.2.3 b) hold and f ′ ∈ FV(Ω)′. Then there is a bounded
+net (f ′
+τ)τ∈T in FV(Ω)′ converging to f ′ in FV(Ω)′
+κ such that Rt
+f(f ′
+τ) ∈ J (E) for
+every τ ∈ T . Due to (13) we obtain that (Rt
+f(f ′
+τ)) is a bounded Cauchy net in the
+quasi-complete space J (E) converging to Rt
+f(f ′) ∈ J (E).
+c) Let Condition 3.2.3 c) hold and f ′ ∈ FV(Ω)′.
+Then there is a sequence
+(f ′
+n)n∈N in FV(Ω)′ converging to f ′ in FV(Ω)′
+κ such that Rt
+f(f ′
+n) ∈ J (E) for every
+n ∈ N. Again (13) implies that (Rt
+f(f ′
+n)) is a Cauchy sequence in the sequentially
+complete space J (E) which converges to Rt
+f(f ′) ∈ J (E).
+d) Let Condition 3.2.3 d) hold and f ′ ∈ FV(Ω)′. Then there is an absolutely
+convex, bounded subset D ⊂ FV(Ω)′
+κ and a sequence (f ′
+n)n∈N in FV(Ω)′ converging
+to f ′ in (FV(Ω)′
+κ)D such that Rt
+f(fn) ∈ J (E) for every n ∈ N. Let r > 0 and
+f ′
+n − f ′
+k ∈ rD. Then Rt
+f(f ′
+n − f ′
+k) ∈ r(Rt
+f(D) ∩ J (E)), implying
+{r > 0 ∣ f ′
+n − f ′
+k ∈ rD} ⊂ {r > 0 ∣ Rt
+f(f ′
+n − f ′
+k) ∈ r(Rt
+f(D) ∩ J (E))
+J (E)}.
+Setting B ∶= Rt
+f(D) ∩ J (E)
+J (E), we derive
+qB(Rt
+f(f ′
+n − f ′
+k)) ≤ qD(f ′
+n − f ′
+k)
+where qB and qD are the gauge functionals of B resp. D. The set Rt
+f(D)∩J (E) is
+absolutely convex as the intersection of two absolutely convex sets and it is bounded
+by (13) and the boundedness of D. So B, being the closure of a disk, is a disk as
+well. Since (f ′
+n) is a Cauchy sequence in (FV(Ω)′
+κ)D, we conclude that (R′
+f(f ′
+n))
+is a Cauchy sequence in J (E)B. The set B is a closed disk in the locally complete
+space J (E) and hence a Banach disk by [89, 10.2.1 Proposition, p. 197]. Thus
+J (E)B is a Banach space and (Rt
+f(f ′
+n)) has a limit g ∈ J (E)B. The continuity
+of the canonical injection J (E)B ↪ J (E) implies that (Rt
+f(f ′
+n)) converges to g in
+J (E) as well. As in a) we obtain that Rt
+f(f ′) = g ∈ J (E).
+e) Let Condition 3.2.3 e) be fulfilled. Let f ∈ FV(Ω,E) and e′ ∈ E′. For every
+f ′ ∈ FV(Ω)′ there are j ∈ J, m ∈ M and C > 0 such that
+∣Rt
+f(f ′)(e′)∣ ≤ C∣Rf(e′)∣j,m =
+(11) C
+sup
+x∈Nj,m(f)
+∣e′(x)∣
+because (T E
+m,T K
+m)m∈M is a strong generator. Since there is K ∈ τ(E) such that
+Nj,m(f) ⊂ K, we have
+∣Rt
+f(f ′)(e′)∣ ≤ C sup
+x∈K
+∣e′(x)∣,
+implying Rt
+f(f ′) ∈ (E′
+τ)′ = J (E) by the Mackey–Arens theorem.
+Therefore we obtain that Rt
+f ∈ L(FV(Ω)′
+κ,J (E)). So we get for all α ∈ A and
+y ∈ F(Ω)′
+pα((J −1 ○ Rt
+f)(y)) =
+(12) pB○α(J ((J −1 ○ Rt
+f)(y))) = pB○α(Rt
+f(y)) ≤
+(13) sup
+x∈Kα
+∣y(x)∣.
+This implies J −1 ○ Rt
+f ∈ L(FV(Ω)′
+κ,E) = FV(Ω)εE (as linear spaces) and we gain
+S(J −1 ○ Rt
+f)(x) = J −1(Rt
+f(δx)) =
+(14) J −1(J (f(x))) = f(x)
+for every x ∈ Ω. Thus S(J −1 ○ Rt
+f) = f, proving the surjectivity of S.
+□
+Further sufficient conditions for S being a topological isomorphism can be found
+in Proposition 5.2.10, Proposition 5.6.6 and Theorem 5.7.1. In particular, we get
+the following corollary as a special case of Theorem 3.2.4.
+
+32
+3. THE ε-PRODUCT FOR WEIGHTED FUNCTION SPACES
+3.2.5. Corollary. Let (T E
+m,T K
+m)m∈M be a strong, consistent generator for
+(FV,E). If
+(i) FV(Ω) is a semi-Montel space and E complete, or
+(ii) FV(Ω) is a Fréchet–Schwartz space and E locally complete, or
+(iii) E is a semi-Montel space, or
+(iv) ∀ f ∈ FV(Ω,E), j ∈ J, m ∈ M ∃ K ∈ κ(E) ∶ Nj,m(f) ⊂ K,
+then FV(Ω) and FV(Ω,E) are ε-compatible, in particluar, FV(Ω,E) ≅ FV(Ω)εE.
+Proof. (i) Follows from Lemma 3.2.2 a) and Theorem 3.2.4 with Condition
+3.2.3 a).
+(ii) If FV(Ω) is a Fréchet–Schwartz space, then we have
+span{δx ∣ x ∈ Ω}
+lc = span{δx ∣ x ∈ Ω}
+FV(Ω)′
+b = span{δx ∣ x ∈ Ω}
+FV(Ω)′
+κ = FV(Ω)′
+by [30, Lemma 6 (b), p. 231] and the bipolar theorem where span{δx ∣ x ∈ Ω}
+lc is the
+local closure of span{δx ∣ x ∈ Ω} in FV(Ω)′
+b. Hence for every f ′ ∈ FV(Ω)′ there is a
+sequence (f ′
+n) in the span of {δx ∣ x ∈ Ω} which converges locally to f ′ in FV(Ω)′
+κ.
+Due to (14) we know that Rt
+f(f ′
+n) ∈ J (E) for every f ∈ FV(Ω,E) and n ∈ N. Since
+Fréchet–Schwartz spaces are also semi-Montel spaces, the statement follows from
+Lemma 3.2.2 a) and Theorem 3.2.4 with Condition 3.2.3 d).
+(iv) Follows from Lemma 3.2.2 c) and Theorem 3.2.4 with Condition 3.2.3 e).
+(iii) Is a special case of (iv) since the set K ∶= acx(Nj,m(f)) is absolutely convex
+and compact in the semi-Montel space E by [89, 6.2.1 Proposition, p. 103] and [89,
+6.7.1 Proposition, p. 112] for every f ∈ FV(Ω,E), j ∈ J and m ∈ M.
+□
+3.2.6. Remark. Linearisations of spaces FV(Ω,E)σ of weak E-valued func-
+tions, where FV(Ω) need not be a dom-space, are treated in [118].
+Let us apply our preceding results to our weighted spaces of k-times continu-
+ously partially differentiable functions on an open set Ω ⊂ Rd with k ∈ N∞.
+3.2.7. Example. Let E be an lcHs, k ∈ N∞, Vk be a directed family of weights
+which is locally bounded away from zero on an open set Ω ⊂ Rd.
+a) CVk(Ω,E) ≅ CVk(Ω)εE if E is a semi-Montel space and CVk(Ω) barrelled.
+b) CVk
+P (∂)(Ω,E) ≅ CVk
+P (∂)(Ω)εE if E is a semi-Montel space and CVk
+P (∂)(Ω)
+barrelled.
+c) CVk(Ω,E) ≅ CVk(Ω)εE if E is complete and CVk(Ω) a Montel space.
+d) CVk
+P (∂)(Ω,E) ≅ CVk
+P (∂)(Ω)εE if E is complete and CVk
+P (∂)(Ω) a Montel
+space.
+e) CVk(Ω,E) ≅ CVk(Ω)εE if E is locally complete and CVk(Ω) a Fréchet–
+Schwartz space.
+f) CVk
+P (∂)(Ω,E) ≅ CVk
+P (∂)(Ω)εE if E is locally complete and CVk
+P (∂)(Ω) a
+Fréchet–Schwartz space.
+Proof. The generator of (CVk,E) and (CVk
+P (∂),E) is strong and consistent
+by Proposition 3.1.10. From Corollary 3.2.5 (iii) we deduce part a) and b), from
+(i) part c) and d) and from (ii) part e) and f).
+□
+Closed subspaces of Fréchet–Schwartz spaces are also Fréchet–Schwartz spaces
+by [131, Proposition 24.18, p. 284]. The space CV∞
+P (∂)(Ω) is closed in CV∞(Ω) if
+there is an lcHs Y such that P(∂)∣CV∞(Ω)∶CV∞(Ω) → Y is continuous. For example,
+this is fulfilled if the coefficients of P(∂) belong to C(Ω), in particular, if P(∂) ∶= ∆
+or ∂, with Y ∶= (C(Ω),τc) due to V∞ being locally bounded away from zero. The
+spaces CVk(Ω) from Example 3.1.9 a)(i) with ωm ∶= Mm × Ω for all m ∈ N0, where
+Mm ∶= {β ∈ Nd
+0 ∣ ∣β∣ ≤ min(m,k)}, are Fréchet spaces and thus barrelled if the J
+
+3.2. ε-COMPATIBILITY
+33
+in Vk ∶= (νj,m)j∈J,m∈N0 is countable by [107, Proposition 3.7, p. 240]. Sufficient
+conditions on the weights that guarantee that CV∞(Ω) is a nuclear Fréchet space
+and hence a Schwartz space as well can be found in [111, Theorem 3.1, p. 188]. For
+the case ωm = Nd
+0 × Ω see the references given in [111, p. 1].
+If Vk = Wk, i.e. Ck(Ω,E) is equipped with its usual topology of uniform conver-
+gence of all partial derivatives up to order k on compact subsets of Ω, Example 3.2.7
+c)+d) can be improved to quasi-complete E. For Ω = Rd this can be found in [158,
+Proposition 9, p. 108, Théorème 1, p. 111] and for general open Ω ⊂ Rd it is already
+mentioned in [94, (9), p. 236] (without a proof) that CWk(Ω,E) ≅ CWk(Ω)εE for
+k ∈ N∞ and quasi-complete E. For k = ∞ we even have CW∞(Ω,E) ≅ CW∞(Ω)εE
+for locally complete E by [30, p. 228]. Our technique allows us to generalise the
+first result and to get back the second result.
+3.2.8. Example. Let E be an lcHs, k ∈ N∞ and Ω ⊂ Rd open. If k < ∞ and E
+has metric ccp, or if k = ∞ and E is locally complete, then
+a) CWk(Ω,E) ≅ CWk(Ω)εE, and
+b) CWk
+P (∂)(Ω,E) ≅ CWk
+P (∂)(Ω)εE if CWk
+P (∂)(Ω) is closed in CWk(Ω).
+Proof. We recall from Example 3.1.9 b) that Wk is the family of weights
+given by νK,m(β,x) ∶= χK(x), (β,x) ∈ Mm × Ω, for all m ∈ N0 and compact K ⊂ Ω
+where Mm ∶= {β ∈ Nd
+0 ∣ ∣β∣ ≤ min(m,k)} and χK is the characteristic function of K.
+We already know that the generator for (CWk,E) and (CWk
+P (∂),E) is strong and
+consistent by Proposition 3.1.10 because Wk is locally bounded away from zero
+on Ω, and CWk(Ω) and its closed subspace CWk
+P (∂)(Ω) are Fréchet spaces. Let
+f ∈ CWk(Ω,E), K ⊂ Ω be compact, m ∈ N0 and consider
+NK,m(f) = {(∂β)Ef(x)νK,m(β,x) ∣ x ∈ Ω, β ∈ Mm} = {0} ∪
+⋃
+β∈Mm
+(∂β)Ef(K).
+NK,m(f) is compact since it is a finite union of compact sets. Furthermore, the
+compact sets {0} and (∂β)Ef(K) are metrisable by [34, Chap. IX, §2.10, Proposi-
+tion 17, p. 159] and thus their finite union NK,m(f) is metrisable as well by [169,
+Theorem 1, p. 361] since the compact set NK,m(f) is collectionwise normal and
+locally countably compact by [63, 5.1.18 Theorem, p. 305]. If E has metric ccp,
+then the set acx(NK,m(f)) is absolutely convex and compact. Thus Corollary 3.2.5
+(iv) settles the case for k < ∞. If k = ∞ and E is locally complete, we observe that
+Kβ ∶= acx((∂β)Ef(K)) for f ∈ CW∞(Ω,E) is absolutely convex and compact by
+[29, Proposition 2, p. 354]. Then we have
+NK,m(f) ⊂ acx( ⋃
+β∈Mm
+Kβ)
+and the set on the right-hand side is absolutely convex and compact by [89, 6.7.3
+Proposition, p. 113]. Again, the statement follows from Corollary 3.2.5 (iv).
+□
+The statement above for k = ∞ follows from Example 3.2.7 e)+f) as well because
+CW∞(Ω) and its closed subspaces are Fréchet–Schwartz spaces. In the context of
+differentiability on infinite dimensional spaces the preceding example a) remains
+true for an open subset Ω of a Fréchet space or DFM-space and quasi-complete
+E by [129, 3.2 Corollary, p. 286].
+Like here this can be generalised to E with
+[metric] ccp. A special case of example b) is already known to be a consequence of
+[30, Theorem 9, p. 232], namely, if k = ∞ and P(∂)K is hypoelliptic with constant
+coefficients. In particular, this covers the space of holomorphic functions and the
+space of harmonic functions. Holomorphy on infinite dimensional spaces is treated
+in [52, Corollary 6.35, p. 332–333] where V = W0, Ω is an open subset of a locally
+
+34
+3. THE ε-PRODUCT FOR WEIGHTED FUNCTION SPACES
+convex Hausdorff k-space and E a quasi-complete locally convex Hausdorff space,
+both over C, which can be generalised to E with [metric] ccp in a similar way.
+For a second improvement of Example 3.2.7 for k = ∞ to locally complete E
+without the condition that CV∞(Ω) resp. CV∞
+P (∂)(Ω) is a Fréchet–Schwartz space
+we introduce the following conditions on the family V∞ on (Mm × Ω)m∈N0. We say
+that a family V∞ of weights on (Mm × Ω)m∈N0 is C1-controlled if
+(i) ∀ j ∈ J, m ∈ N0, β ∈ Mm ∶ νj,m(β,⋅) ∈ C1(Ω),
+(ii) ∀ j ∈ J, m ∈ N0, β,γ ∈ Mm,x ∈ Ω ∶ νj,m(β,x) = νj,m(γ,x),
+(iii) ∀ j ∈ J, m ∈ N0 ∃ i ∈ J, k ∈ N0, k ≥ m, C > 0 ∀ β ∈ Mm, x ∈ Ω, 1 ≤ n ≤ d ∶
+∣∂enνj,m(β,⋅)∣(x) ≤ Cνi,k(β,x).
+We say that family Vk, k ∈ N∞, fulfils condition (V∞) if
+∀ m ∈ N0, j ∈ J ∃ n ∈ N≥m, i ∈ J ∀ ε > 0 ∃ K ⊂ Ω compact ∀ β ∈ Mm, x ∈ Ω ∖ K ∶
+νj,m(β,x) ≤ ενi,n(β,x)
+where N≥m ∶= {n ∈ N0 ∣ n ≥ m}. Here (V∞) stands for vanishing at infinity and
+the condition was introduced in [107, Remark 3.4, p. 239] and for k = 0 in [16, 1.3
+Bemerkung, p. 189].
+3.2.9. Example. Let E be an lcHs and V∞ a directed C1-controlled family of
+weights on an open convex set Ω ⊂ Rd which fulfils (V∞). If E is locally complete,
+then
+a) CV∞(Ω,E) ≅ CV∞(Ω)εE if CV∞(Ω) is barrelled, and
+b) CV∞
+P (∂)(Ω,E) ≅ CV∞
+P (∂)(Ω)εE if CV∞
+P (∂)(Ω) is barrelled.
+Proof. We already know that the generator for (CV∞,E) and (CV∞
+P (∂),E) is
+strong and consistent by Proposition 3.1.10 because V∞ is locally bounded away
+from zero on Ω as νj,m(β,⋅) is continuous for all j ∈ J, m ∈ N0 and β ∈ Mm.
+Let f ∈ CV∞(Ω,E), j ∈ J, m ∈ N0 and β ∈ Mm. We set g∶Ω → E, g(x) ∶=
+(∂β)Ef(x)νj,m(β,x), and note that
+(∂en)Eg(x) = (∂β+en)Ef(x)νj,m(β,x) + (∂β)Ef(x)((∂en)Rνj,m(β,⋅))(x),
+x ∈ Ω,
+for all 1 ≤ n ≤ d.
+Since V∞ is directed and C1-controlled there are i1,i2 ∈ J,
+k1,k2 ∈ N0, k1 > m, k2 ≥ m, and C1,C2 > 0 such that
+pα((∂en)Eg(x))
+≤ pα((∂β+en)Ef(x))νj,m(β,x) + pα((∂β)Ef(x))∣(∂en)Rνj,m(β,⋅)∣(x)
+≤ C1pα((∂β+en)Ef(x))νi1,k1(β,x) + C2pα((∂β)Ef(x))νi2,k2(β,x)
+= C1pα((∂β+en)Ef(x))νi1,k1(β + en,x) + C2pα((∂β)Ef(x))νi2,k2(β,x)
+for all 1 ≤ n ≤ d and α ∈ A, which implies
+sup
+x∈Ω
+γ∈Nd
+0,∣γ∣≤1
+pα((∂γ)Eg(x)) ≤ ∣f∣j,m,α + C1∣f∣i1,k1,α + C2∣f∣i2,k2,α.
+Thus g is (weakly) C1
+b .
+Due to (V∞) there are n ∈ N≥m and i ∈ J such that for all ε > 0 there is a
+compact set K ⊂ Ω such that for all β ∈ Mm and x ∈ Ω ∖ K we have
+νj,m(β,x) ≤ ενi,n(β,x).
+Since V∞ is directed, we may assume w.l.o.g. that νj,m(β,x) ≤ νi,n(β,x) for all
+x ∈ Ω. This implies that the zeros of νi,n(β,⋅) are zeros of νj,m(β,⋅). We define
+h∶Ω → [0,∞) by h(x) ∶= νi,n(β,x)/νj,m(β,x) for x ∈ Ω with νj,m(β,x) ≠ 0 and
+
+3.2. ε-COMPATIBILITY
+35
+h(x) ∶= 1 if νj,m(β,x) = 0. We note that h(x) > 0 for all x ∈ Ω as the zeros of
+νi,n(β,⋅) are contained in the zeros of νj,m(β,⋅). It follows that
+(∂β)Ef(x)νj,m(β,x)h(x) = (∂β)Ef(x)νi,n(β,x)
+for x ∈ Ω with νj,m(β,x) ≠ 0 and (∂β)Ef(x)νj,m(β,x)h(x) = 0 for x ∈ Ω with
+νj,m(β,x) = 0. Therefore (∂β)Efνj,m(β,⋅)h is bounded on Ω. Further,
+εh(x) = ενi,n(β,x)/νj,m(β,x) ≥ 1
+for x ∈ Ω ∖ K with νj,m(β,x) ≠ 0 because (V∞) is fulfilled. Further, the zeros of
+νj,m(β,⋅) are contained in N ∶= {x ∈ Ω ∣ (∂β)Ef(x)νj,m(β,x) = 0}. This yields that
+Kβ ∶= acx((∂β)Efνj,m(β,⋅)(Ω)) is absolutely convex and compact by Proposition
+A.1.4 and A.1.5. Furthermore,
+Nj,m(f) = {(∂β)Ef(x)νj,m(β,x) ∣ x ∈ Ω, β ∈ Mm} ⊂ acx( ⋃
+β∈Mm
+Kβ)
+and the set on the right-hand side is absolutely convex and compact by [89, 6.7.3
+Proposition, p. 113]. Finally, our statement follows from Corollary 3.2.5 (iv).
+□
+For the Schwartz space S(Rd,E) and the multiplier space OM(Rd,E) from
+Example 3.1.9 c) and d) an improvement of Example 3.2.7 c) to quasi-complete E
+is already known, see e.g. [158, Proposition 9, p. 108, Théorème 1, p. 111]. However,
+due to Example 3.2.9 it is even allowed that E is only locally complete.
+3.2.10. Corollary. If E is a locally complete lcHs, then S(Rd,E) ≅ S(Rd)εE
+and OM(Rd,E) ≅ OM(Rd)εE.
+Proof. We start with the Schwartz space. Due to Example 3.2.9 a) and the
+barrelledness of the Fréchet space S(Rd) we only need to check that its directed
+family V∞ ∶= (ν1,m)m∈N0 of weights given by ν1,m(β,x) ∶= (1 + ∣x∣2)m/2, x ∈ Rd, for
+m ∈ N0 and β ∈ Mm is C1-controlled and fulfils (V∞). Obviously, condition (i) and
+(ii) are fulfilled. Since
+∣∂enν1,m(β,⋅)∣(x) = (m/2)(1 + ∣x∣2)(m/2)−12∣xn∣ ≤ m(1 + ∣x∣2)m/2 = mν1,m(β,x)
+for all x ∈ Rd and 1 ≤ n ≤ d, condition (iii) is also fulfilled. Thus V∞ is C1-controlled.
+Noting that for every m ∈ N and ε > 0 there is r > 0 such that
+(1 + ∣x∣2)m/2
+(1 + ∣x∣2)m = (1 + ∣x∣2)−m/2 ≤ ε
+for all x ∉ Br(0), we obtain that
+ν1,m(β,x) ≤ εν1,2m(β,x)
+for all x ∉ Br(0) and β ∈ Mm. Hence V∞ fulfils condition (V∞).
+Now, let us consider the multiplier space. We already know that the generator
+for (OM,E) is strong and consistent by Proposition 3.1.10 because OM(R) is a
+Montel space, thus barrelled, by [83, Chap. II, §4, n○4, Théorème 16, p. 131] and
+its family of weights is continuous on Rd, thus locally bounded away from zero.
+Let f ∈ OM(R,E), g ∈ S(Rd), m ∈ N0 and β ∈ Mm. Then (∂β)Ef ∈ OM(R,E)
+and hence ((∂β)Ef)g ∈ S(Rd,E), which implies that ((∂β)Ef)g ∈ C1
+b (Rd,E). More-
+over, we choose h∶Rd → (0,∞), h(x) ∶= 1 + ∣x∣2. Then ((∂β)Ef)gh is bounded on
+Rd and for ε > 0 there is r > 0 such that (1 + ∣x∣2)−1 ≤ ε for all x ∉ Br(0), yielding
+that Kβ,g ∶= acx(((∂β)Ef)g(Rd)) is absolutely convex and compact by Proposi-
+tion A.1.4 and A.1.5.
+Let j ⊂ S(Rd) be finite.
+Since for each x ∈ Rd we have
+(∂β)Ef(x)maxg∈j ∣g(x)∣ = eiθ(∂β)Ef(x)̃g(x) for some ̃g ∈ j and θ ∈ [0,2π), we get
+Nj,m(f) = {(∂β)Ef(x)max
+g∈j ∣g(x)∣ ∣ x ∈ Rd, β ∈ Mm} ⊂ acx(
+⋃
+β∈Mm,g∈j
+Kβ,g).
+
+36
+3. THE ε-PRODUCT FOR WEIGHTED FUNCTION SPACES
+The set on the right-hand side is absolutely convex and compact by [89, 6.7.3
+Proposition, p. 113]. Finally, our statement follows from Corollary 3.2.5 (iv).
+□
+For an alternative proof in the case of the Schwartz space we may also use
+Example 3.2.7 e) since S(Rd) is a Fréchet–Schwartz space.
+Example 3.2.9 can
+also be used for an alternative proof of Example 3.2.8 if k = ∞ by observing that
+CW∞(Ω,E) = CV∞(Ω,E) for any lcHs E where V∞ ∶= {ν ∈ C∞
+c (Ω) ∣ ν ≥ 0} and
+C∞
+c (Ω) is the space of functions in C∞(Ω) with compact support.
+Now, we improve Example 3.2.7 for the special case of spaces of ultradifferen-
+tiable functions E(Mp)(Ω,E) and E{Mp}(Ω,E) from Example 3.1.9 e) and f) where
+ωm ∶= Nd
+0 × Ω for all m ∈ N0. For this purpose we recall the following conditions of
+Komatsu for the sequence (Mp)p∈N0 (see [99, p. 26] and [101, p. 653]):
+(M.0) M0 = M1 = 1,
+(M.1) ∀ p ∈ N ∶ M 2
+p ≤ Mp−1Mp+1,
+(M.2)’ ∃ A,C > 0 ∀ p ∈ N0 ∶ Mp+1 ≤ ACp+1Mp,
+(M.3)’ ∑∞
+p=1
+Mp−1
+Mp < ∞.
+3.2.11. Example. Let E be an lcHs, Ω ⊂ Rd open and (Mp)p∈N0 a sequence of
+positive real numbers.
+a) E(Mp)(Ω,E) ≅ E(Mp)(Ω)εE if E is locally complete.
+b) E{Mp}(Ω,E) ≅ E{Mp}(Ω)εE if E is complete or semi-Montel and in both
+cases (Mp)p∈N0 fulfils (M.1) and (M.3)’.
+c) E{Mp}(Ω,E) ≅ E{Mp}(Ω)εE if E is sequentially complete and (Mp)p∈N0
+fulfils (M.0), (M.1), (M.2)’ and (M.3)���.
+Proof. The generator is strong and consistent by Proposition 3.1.10 since the
+family of weights given in Example 3.1.9 e) resp. f) is locally bounded away from
+zero on Ω and E(Mp)(Ω) is a Fréchet–Schwartz space in a) by [99, Theorem 2.6,
+p. 44] whereas E{Mp}(Ω) is a Montel space in b) and c) by [99, Theorem 5.12, p.
+65–66]. Hence the statements a) and b) follow from Example 3.2.7.
+Let us turn to c). We note that E{Mp}(Ω,E) ⊂ E{Mp}(Ω,E)κ by Lemma 3.2.2
+a) for any lcHs E.
+Further, we claim that Condition 3.2.3 c) is fulfilled.
+Let
+f ′ ∈ E{Mp}(Ω)′. Due to [101, Proposition 3.7, p. 677] there is a sequence (fn)n∈N in
+the space D{Mp}(Ω) of ultradifferentiable functions of class {Mp} of Roumieu-type
+with compact support which converges to f ′ in E{Mp}(Ω)′
+b. Let f ∈ E{Mp}(Ω,E).
+We observe that for every e′ ∈ E′
+∣Rt
+f(fn)(e′)∣ = ∣∫
+Ω
+fn(x)e′(f(x))dx∣ ≤ λ(supp(fn))
+sup
+y∈Kn(f)
+∣e′(y)∣
+where λ is the Lebesgue measure, supp(fn) is the support of fn and Kn(f) ∶=
+{fn(x)f(x) ∣ x ∈ supp(fn)}. The set Kn(f) is compact and metrisable by [34,
+Chap. IX, §2.10, Proposition 17, p. 159] and thus the closure of its absolutely
+convex hull is compact in E as the sequentially complete space E has metric ccp.
+We conclude that Rt
+f(fn) ∈ (E′
+κ)′ = J (E) for every n ∈ N. Therefore Condition
+3.2.3 c) is fulfilled, implying statement c) for sequentially complete E by Theorem
+3.2.4.
+□
+The results a) and b) in this example are new whereas c) is already proved in
+[101, Theorem 3.10, p. 678] in a different way. In particular, part a) improves [101,
+Theorem 3.10, p. 678] since Komatsu’s conditions (M.0), (M.1), (M.2)’ and (M.3)’
+are not needed and the condition that E is sequentially complete is weakened to
+local completeness. We included c) to demonstrate an application of Condition
+3.2.3 c).
+
+CHAPTER 4
+Consistency
+4.1. The spaces AP(Ω,E) and consistency
+This section is dedicated to the properties of functions which are compatible
+with the ε-product in the sense that the space of functions having these properties
+can be chosen as the space AP(Ω,E) or ⋂m∈M domT E
+m in the Definition 3.1.7 b) of
+consistency. This is done in a quite general way so that we are not tied to certain
+spaces and have to redo our argumentation, for example, if we consider the same
+generator (T E
+m,T K
+m)m∈M for two different spaces of functions.
+Due to the linearity and continuity of u ∈ FV(Ω)εE for a dom-space FV(Ω)
+and S(u) = u ○ δ with δ∶Ω → FV(Ω)′, x ↦ δx, these are properties which are purely
+pointwise or given by pointwise approximation. Among such properties of func-
+tions are continuity by Proposition 4.1.1, Cauchy continuity by Proposition 4.1.3,
+uniform continuity by Proposition 4.1.5, continuous extendability by Proposition
+4.1.7, continuous differentiability by Proposition 3.1.10, vanishing at infinity by
+Proposition 4.1.9 and purely pointwise properties of a function like vanishing on a
+set by Proposition 4.1.10.
+We collect these properties in propositions and in follow-up lemmas we handle
+properties which can be described by compositions of defining operators T E
+m1 ○ T E
+m2
+like continuous differentiability (of higher order) of Fourier transformations (see
+Example 4.2.26). We fix the following notation for this section. For a dom-space
+FV(Ω) and linear T∶FV(Ω) → KΩ we set (δ ○ T)(x)(f) ∶= (δx ○ T)(f) ∶= T(f)(x)
+for all x ∈ Ω and f ∈ FV(Ω).
+4.1.1. Proposition (continuity). Let Ω be a topological Hausdorff space and
+FV(Ω) a dom-space such that FV(Ω) ⊂ C(Ω) as a linear subspace. Then S(u) ∈
+C(Ω,E) for all u ∈ FV(Ω)εE if δ ∈ C(Ω,FV(Ω)′
+κ).
+Proof. Let u ∈ FV(Ω)εE. Since S(u) = u○δ and δ ∈ C(Ω,FV(Ω)′
+κ), we obtain
+that S(u) is in C(Ω,E).
+□
+Now, we tackle the problem of the continuity of δ∶Ω → FV(Ω)′
+κ in the proposi-
+tion above and phrase our solution in a way such that it can be applied to show the
+continuity of the partial derivative (∂β)E(S(u)) as well (see Proposition 3.1.11).
+We recall that a topological space Ω is called completely regular if for any non-empty
+closed subset A ⊂ Ω and x ∈ Ω ∖ A there is f ∈ C(Ω,[0,1]) such that f(x) = 0 and
+f(z) = 1 for all z ∈ A (see [88, Definition 11.1, p. 180]). Examples of completely
+regular spaces are uniformisable, particularly metrisable, spaces by [88, Proposition
+11.5, p. 181] and locally convex Hausdorff spaces by [65, Proposition 3.27, p. 95].
+A completely regular space Ω is a kR-space if for any completely regular space Y
+and any map f∶Ω → Y , whose restriction to each compact K ⊂ Ω is continuous,
+the map is already continuous on Ω (see [37, (2.3.7) Proposition, p. 22]). Examples
+of kR-spaces are completely regular k-spaces by [63, 3.3.21 Theorem, p. 152]. A
+topological space Ω is called k-space (compactly generated space) if it satisfies the
+following condition: A ⊂ Ω is closed if and only if A ∩ K is closed in K for every
+compact K ⊂ Ω. Every locally compact Hausdorff space is a completely regular
+37
+
+38
+4. CONSISTENCY
+k-space. Further, every sequential Hausdorff space is a k-space by [63, 3.3.20 The-
+orem, p. 152], in particular, every first-countable Hausdorff space. Thus metrisable
+spaces are completely regular Hausdorff k-spaces. Moreover, the dual space (X′,τc)
+with the topology of compact convergence τc is an example of a completely regular
+Hausdorff k-space that is neither locally compact nor metrisable by [178, p. 267] if
+X is an infinite-dimensional Fréchet space.
+We denote by CW(Ω) the space of scalar-valued continuous functions on a
+topological Hausdorff space Ω with the topology τc of compact convergence, i.e. the
+topology of uniform convergence on compact subsets, which itself is the weighted
+topology given by the family of weights W ∶= W0 ∶= {χK ∣ K ⊂ Ω compact}, and
+by Cb(Ω) the space of scalar-valued bounded, continuous functions on Ω with the
+topology of uniform convergence on Ω.
+4.1.2. Lemma. Let Ω be a topological Hausdorff space, FV(Ω) a dom-space
+and T∶FV(Ω) → C(Ω) linear. Then δ ○T ∈ C(Ω,FV(Ω)′
+γ) in each of the subsequent
+cases:
+(i) Ω is a kR-space and T∶FV(Ω) → CW(Ω) is continuous.
+(ii) T∶FV(Ω) → Cb(Ω) is continuous.
+Proof. First, if x ∈ Ω and (xτ)τ∈T is a net in Ω converging to x, then we
+observe that
+(δxτ ○ T)(f) = T(f)(xτ) → T(f)(x) = (δx ○ T)(f)
+for every f ∈ FV(Ω) as T(f) is continuous on Ω.
+(i) Verbatim as in Proposition 3.1.11 a).
+(ii) There are j ∈ J, m ∈ M and C > 0 such that
+sup
+x∈Ω
+∣(δx ○ T)(f)∣ = sup
+x∈Ω
+∣T(f)(x)∣ ≤ C∣f∣FV(Ω),j,m
+for every f ∈ FV(Ω). This means that {δx ○T ∣ x ∈ Ω} is equicontinuous in FV(Ω)′,
+yielding the statement like before.
+□
+The preceding lemma is just a modification of [16, 4.1 Lemma, p. 198] where
+FV(Ω) = CV(Ω), the Nachbin-weighted space of continuous functions, and T = id.
+Next, we turn to Cauchy continuity. A function f∶Ω → E from a metric space
+Ω to an lcHs E is called Cauchy continuous if it maps Cauchy sequences to Cauchy
+sequences. We write CC(Ω,E) for the space of Cauchy continuous functions from
+Ω to E and set CC(Ω) ∶= CC(Ω,K).
+4.1.3. Proposition (Cauchy continuity). Let Ω be a metric space and FV(Ω)
+a dom-space such that FV(Ω) ⊂ CC(Ω) as a linear subspace. Then S(u) ∈ CC(Ω,E)
+for all u ∈ FV(Ω)εE if δ ∈ CC(Ω,FV(Ω)′
+κ).
+Proof. Let u ∈ FV(Ω)εE and (xn) a Cauchy sequence in Ω. Then (δxn) is a
+Cauchy sequence in FV(Ω)′
+κ since δ ∈ CC(Ω,FV(Ω)′
+κ). It follows that (S(u)(xn)) is
+a Cauchy sequence in E because u is uniformly continuous and u(δxn) = S(u)(xn).
+Hence we conclude that S(u) ∈ CC(Ω,E).
+□
+For the next lemma we equip the space CC(Ω) with the topology of uniform
+convergence on precompact subsets of Ω.
+4.1.4. Lemma. Let FV(Ω) be a dom-space and T ∈ L(FV(Ω),CC(Ω)) for a
+metric space Ω. Then δ ○ T ∈ CC(Ω,FV(Ω)′
+γ).
+Proof. Let (xn) be a Cauchy sequence in Ω. We have (δxn○T)(f) = T(f)(xn)
+for every f ∈ FV(Ω), which implies that ((δxn ○ T)(f)) is a Cauchy sequence in
+K because T(f) ∈ CC(Ω) by assumption.
+Since K is complete, it has a unique
+limit T∞(f) ∶= limn→∞(δxn ○ T)(f) defining a linear functional in f. The set N ∶=
+
+4.1. THE SPACES AP(Ω, E) AND CONSISTENCY
+39
+{xn ∣ n ∈ N} is precompact in Ω since Cauchy sequences are precompact. Hence
+there are j ∈ J, m ∈ M and C > 0 such that
+sup
+n∈N
+∣(δxn ○ T)(f)∣ = sup
+x∈N
+∣T(f)(x)∣ ≤ C∣f∣FV(Ω),j,m
+for every f ∈ FV(Ω). Therefore the set {δxn○T ∣ n ∈ N} is equicontinuous in FV(Ω)′,
+which implies that T∞ ∈ FV(Ω)′ and the convergence of (δxn ○T) to T∞ in FV(Ω)′
+γ
+due to the observation in the beginning and the fact that γ(FV(Ω)′,FV(Ω)) and
+σ(FV(Ω)′,FV(Ω)) coincide on equicontinuous sets. In particular, (δxn ○ T) is a
+Cauchy sequence in FV(Ω)′
+γ. Furthermore, for every x ∈ Ω we obtain from the
+choice xn = x for all n ∈ N that δx ○ T ∈ FV(Ω)′. Thus the map δ ○ T∶Ω → FV(Ω)′
+γ
+is well-defined and Cauchy continuous.
+□
+The subsequent proposition and lemma handle the analogous statements for
+uniform continuity.
+For a metric space Ω we denote by Cu(Ω,E) the space of
+uniformly continuous functions from Ω to E and set Cu(Ω) ∶= Cu(Ω,K).
+4.1.5. Proposition (uniform continuity). Let (Ω,d) be a metric space and
+FV(Ω) a dom-space such that FV(Ω) ⊂ Cu(Ω) as a linear subspace. Then S(u) ∈
+Cu(Ω,E) for all u ∈ FV(Ω)εE if δ ∈ Cu(Ω,FV(Ω)′
+κ).1
+Proof. Let (zn), (xn) be sequences in Ω with limn→∞ d(zn,xn) = 0 and u ∈
+FV(Ω)εE. Then (δzn −δxn) converges to 0 in FV(Ω)′
+κ because δ ∈ Cu(Ω,FV(Ω)′
+κ).
+As a consequence (S(u)(zn) − S(u)(xn)) converges to 0 in E since u is uniformly
+continuous and u(δzn −δxn) = S(u)(zn)−S(u)(xn). Hence we conclude that S(u) ∈
+Cu(Ω,E).
+□
+For the next lemma we mean by Cbu(Ω) the space of scalar-valued bounded,
+uniformly continuous functions equipped with the topology of uniform convergence
+on a metric space Ω.
+4.1.6. Lemma. Let FV(Ω) be a dom-space and T ∈ L(FV(Ω),Cbu(Ω)) for a
+metric space (Ω,d). Then δ ○ T ∈ Cu(Ω,FV(Ω)′
+γ).
+Proof. Let (zn) and (xn) be sequences in Ω such that limn→∞ d(zn,xn) = 0.
+We have
+(δzn ○ T − δxn ○ T)(f) = T(f)(zn) − T(f)(xn)
+for every f ∈ FV(Ω), which implies that (δzn ○ T − δxn ○ T)(f) converges to 0 in K
+for every f ∈ FV(Ω) because T(f) ∈ Cu(Ω). There exist j ∈ J, m ∈ M and C > 0
+such that
+sup
+n∈N
+∣(δzn ○ T − δxn ○ T)(f)∣ ≤ 2sup
+x∈Ω
+∣T(f)(x)∣ ≤ 2C∣f∣FV(Ω),j,m
+for every f ∈ FV(Ω). Therefore the set {δzn ○ T − δxn ○ T ∣ n ∈ N} is equicontinuous
+in FV(Ω)′ and we conclude the statement like before.
+□
+Let us turn to continuous extensions. Let X be a metric space and Ω ⊂ X. We
+write Cext(Ω,E) for the space of functions f ∈ C(Ω,E) which have a continuous
+extension to Ω and set Cext(Ω) ∶= Cext(Ω,K).
+4.1.7. Proposition (continuous extendability). Let X be a metric space, Ω ⊂ X
+and FV(Ω) a dom-space such that FV(Ω) ⊂ Cext(Ω) as a linear subspace. Then
+S(u) ∈ Cext(Ω,E) for all u ∈ FV(Ω)εE if δ ∈ Cext(Ω,FV(Ω)′
+κ).
+1Here, we use the symbol u for elements in FV(Ω)εE instead of the usual u to avoid confusion
+with the index u of Cu(Ω) resp. Cu(Ω, E).
+
+40
+4. CONSISTENCY
+Proof. Let u ∈ FV(Ω)εE. There is δext ∈ C(Ω,FV(Ω)′
+κ) such that δext = δ on
+Ω since δ ∈ Cext(Ω,FV(Ω)′
+κ). Moreover, u○δext ∈ C(Ω,E) and equal to S(u) = u○δ
+on Ω, yielding S(u) ∈ Cext(Ω,E).
+□
+For the next lemma we equip Cext(Ω) with the topology of uniform convergence
+on compact subsets of Ω.
+4.1.8. Lemma. Let X be a metric space, Ω ⊂ X, FV(Ω) a dom-space and
+T ∈ L(FV(Ω),Cext(Ω)). Then δ ○ T ∈ Cext(Ω,FV(Ω)′
+γ) if FV(Ω) is barrelled.
+Proof. From Lemma 4.1.2 (i) we derive that δ○T ∈ C(Ω,FV(Ω)′
+γ). Let x ∈ ∂Ω
+and (xn) be a sequence in Ω with xn → x. Then (δxn ○ T) is a sequence in FV(Ω)′
+and
+lim
+n→∞(δxn ○ T)(f) = lim
+n→∞T(f)(xn) =∶ (δext
+x
+○ T)(f)
+in K for every f ∈ FV(Ω), which implies that (δxn○T) converges to δext
+x
+○T pointwise
+on FV(Ω) because T(f) ∈ Cext(Ω). As a consequence of the Banach–Steinhaus
+theorem we get (δext
+x
+○ T) ∈ FV(Ω)′ and the convergence in FV(Ω)′
+γ.
+□
+Let FV(Ω,E) be a dom-space, X a set, K a family of sets and π∶⋃m∈M ωm → X
+such that ⋃K∈K K ⊂ X.
+We say that a function f ∈ ⋂m∈M domT E
+m vanishes at
+infinity in the weighted topology w.r.t. (π,K) if
+∀ ε > 0, j ∈ J, m ∈ M, α ∈ A ∃ K ∈ K ∶
+sup
+x∈ωm,
+π(x)∉K
+pα(T E
+m(f)(x))νj,m(x) < ε.
+(16)
+Further, we set
+APπ,K(Ω,E) ∶= {f ∈ ⋂
+m∈M
+domT E
+m ∣ f fulfils (16)}.
+4.1.9. Proposition (vanishing at ∞ w.r.t. to (π,K)). Let (T E
+m,T K
+m)m∈M be
+the generator for (FV,E), let FV(Ω,Y ) ⊂ APπ,K(Ω,Y ) as a linear subspace for
+Y ∈ {K,E} and K be closed under taking finite unions.
+(i) If for all u ∈ FV(Ω)εE it holds that S(u) ∈ ⋂m∈M dom(T E
+m) and
+∀ m ∈ M, x ∈ ωm ∶ (T E
+mS(u))(x) = u(T K
+m,x),
+(17)
+then S(u) ∈ APπ,K(Ω,E) for all u ∈ FV(Ω)εE.
+(ii) If for all e′ ∈ E′ and f ∈ FV(Ω,E) it holds that e′ ○ f ∈ ⋂m∈M dom(T K
+m)
+and
+∀ m ∈ M, x ∈ ωm ∶ T K
+m(e′ ○ f)(x) = (e′ ○ T E
+m(f))(x),
+(18)
+then e′ ○ f ∈ APπ,K(Ω) for all e′ ∈ E′ and f ∈ FV(Ω,E).
+Proof. (i) We set Bj,m ∶= {f ∈ FV(Ω) ∣ ∣f∣j,m ≤ 1} for j ∈ J and m ∈ M. Let
+u ∈ FV(Ω)εE. The topologies σ(FV(Ω)′,FV(Ω)) and κ(FV(Ω)′,FV(Ω)) coincide
+on the equicontinuous set B○
+j,m and we deduce that the restriction of u to B○
+j,m is
+σ(FV(Ω)′,FV(Ω))-continuous.
+Let ε > 0, j ∈ J, m ∈ M, α ∈ A and set Uα,ε ∶= {x ∈ E ∣ pα(x) < ε}. Then there
+are a finite set N ⊂ FV(Ω) and η > 0 such that u(f ′) ∈ Uα,ε for all f ′ ∈ VN,η where
+VN,η ∶= {f ′ ∈ FV(Ω)′ ∣ sup
+f∈N
+∣f ′(f)∣ < η} ∩ B○
+j,m
+because the restriction of u to B○
+j,m is σ(FV(Ω)′,FV(Ω))-continuous. Since N ⊂
+FV(Ω) is finite, FV(Ω) ⊂ APπ,K(Ω) and K is closed under taking finite unions,
+there is K ∈ K such that
+sup
+x∈ωm
+π(x)∉K
+∣T K
+m(f)(x)∣νj,m(x) < η
+(19)
+
+4.2. FURTHER EXAMPLES OF ε-PRODUCTS
+41
+for every f ∈ N. It follows from (19) and (the proof of) Lemma 3.1.8 that
+Dπ⊄K,j,m ∶= {T K
+m,x(⋅)νj,m(x) ∣ x ∈ ωm, π(x) ∉ K} ⊂ VN,η
+and thus u(Dπ⊄K,j,m) ⊂ Uα,ε. Therefore we have
+sup
+x∈ωm
+π(x)∉K
+pα(T E
+m(S(u))(x))νj,m(x) =
+(17)
+sup
+x∈ωm
+π(x)∉K
+pα(u(T K
+m,x))νj,m(x) < ε.
+Hence we conclude that S(u) ∈ APπ,K(Ω,E).
+(ii) Let ε > 0, f ∈ FV(Ω,E) and e′ ∈ E′. Then there exist α ∈ A and C > 0 such
+that ∣e′(x)∣ ≤ Cpα(x) for every x ∈ E. For j ∈ J and m ∈ M there is K ∈ K such that
+sup
+x∈ωm
+π(x)∉K
+pα(T E
+m(f)(x))νj,m(x) < ε
+C
+since FV(Ω,E) ⊂ APπ,K(Ω,E). It follows that
+sup
+x∈ωm
+π(x)∉K
+∣T K
+m(e′ ○ f)(x)∣νj,m(x) =
+(18)
+sup
+x∈ωm
+π(x)∉K
+∣e′(T E
+m(f)(x))∣νj,m(x) < C ε
+C = ε,
+yielding e′ ○ f ∈ APπ,K(Ω).
+□
+The first part of the proof above adapts an idea in the proof of [16, 4.4 Theo-
+rem, p. 199–200] where (T E
+m,T K
+m)m∈M = (idEΩ,idKΩ) which is a special case of our
+proposition.
+Our last proposition of this section is immediate. For ω ⊂ Ω we set APω(Ω,E) ∶=
+{f ∈ EΩ ∣ ∀ x ∈ ω ∶ f(x) = 0} and APω(Ω) ∶= APω(Ω,K).
+4.1.10. Proposition (vanishing on a subset). Let ω ⊂ Ω and FV(Ω) a dom-
+space such that FV(Ω) ⊂ APω(Ω) as a linear subspace. Then S(u) ∈ APω(Ω,E)
+for all u ∈ FV(Ω)εE.
+4.2. Further examples of ε-products
+In Chapter 3 we dealt with weighted spaces of continuously partially differen-
+tiable functions. Now, we treat many examples of weighted spaces FV(Ω,E) of
+functions with less regularity on a set Ω with values in a locally convex Hausdorff
+space E over the field K. Applying the results of the preceding sections, we give
+conditions on E such that FV(Ω) and FV(Ω,E) are ε-compatible, in particular,
+that
+FV(Ω,E) ≅ FV(Ω)εE
+holds. We start with the simplest example of all. Let Ω be a non-empty set and
+equip the space EΩ with the topology of pointwise convergence, i.e. the locally
+convex topology given by the seminorms
+∣f∣K,α ∶= sup
+x∈K
+pα(f(x))χK(x),
+f ∈ EΩ,
+for finite K ⊂ Ω and α ∈ A. To prove EN0 ≅ KN0εE for complete E is given as an
+exercise in [94, Aufgabe 10.5, p. 259], which we generalise now.
+4.2.1. Example. Let Ω be a non-empty set and E an lcHs. Then EΩ ≅ KΩεE.
+Proof. The strength and consistency of the generator (idEΩ,idKΩ) is obvious.
+Let f ∈ EΩ, K ⊂ Ω be finite and set NK(f) ∶= f(Ω)χK(Ω). Then we have NK(f) =
+f(K) ∪ {0} if K ≠ Ω, and NK(f) = f(K) if K = Ω. Thus NK(f) is finite, hence
+compact, NK(f) ⊂ acx(f(K)) and acx(f(K)) is a subset of the finite dimensional
+subspace span(f(K)) of E. It follows that acx(f(K)) is compact by [89, 6.7.4
+Proposition, p. 113], implying our statement by virtue of Corollary 3.2.5 (iv).
+□
+
+42
+4. CONSISTENCY
+The next example will give us the counterpart of Example 3.2.9 a) on the level
+of sequence spaces. Let Ω be a set, E an lcHs and V ∶= (νj)j∈J a directed family of
+weights νj∶Ω → [0,∞) on Ω. We set
+ℓV(Ω,E) ∶= {f ∈ EΩ ∣ ∀ j ∈ J, α ∈ A ∶ ∣f∣j,α ∶= sup
+x∈Ω
+pα(f(x))νj(x) < ∞}
+and ℓV(Ω) ∶= ℓV(Ω,K).
+4.2.2. Example. Let E be an lcHs, (Ω,d) a uniformly discrete metric space,
+i.e. there is r > 0 such that d(x,y) ≥ r for all x,y ∈ Ω, x ≠ y, and V ∶= (νj)j∈J a
+directed family of weights on Ω such that
+∀ j ∈ J ∃ i ∈ J ∀ ε > 0 ∃ K ⊂ Ω compact ∀ x ∈ Ω ∖ K ∶ νj(x) ≤ ενi(x).
+(20)
+If E is locally complete, then ℓV(Ω,E) ≅ ℓV(Ω)εE.
+Proof. Let f ∈ ℓV(Ω,E) and j ∈ J. Then fνj is bounded on Ω by definition
+of ℓV(Ω,E). Since (Ω,d) is uniformly discrete, there is r > 0 such that
+pα(f(x)νj(x) − f(y)νj(y))
+d(x,y)
+≤ 2
+r ∣f∣j,α < ∞,
+x,y ∈ Ω, x ≠ y,
+for every α ∈ A. Therefore fνj ∈ C[1]
+b (Ω,E) where
+C[1]
+b (Ω,E) ∶= {g ∈ EΩ ∣ ∀α ∈ A ∶ sup
+x∈Ω
+pα(g(x)) < ∞ and sup
+x,y∈Ω
+x≠y
+pα(g(x) − g(y))
+d(x,y)
+< ∞}.
+Due to (20) there is i ∈ J such that for all ε > 0 there exists a compact set K ⊂ Ω
+such that νj(x) ≤ ενi(x) for all x ∈ Ω∖K. As V is directed, we may assume w.l.o.g.
+that νj(x) ≤ νi(x) for all x ∈ Ω. This implies that the zeros of νi are zeros of νj. We
+define h∶Ω → [0,∞) by h(x) ∶= νi(x)/νj(x) for x ∈ Ω with νj(x) ≠ 0 and h(x) ∶= 1
+if νj(x) = 0. We observe that h(x) > 0 for all x ∈ Ω as the zeros of νi are contained
+in the zeros of νj. It follows that
+f(x)νj(x)h(x) = f(x)νi(x)
+for x ∈ Ω with νj(x) ≠ 0 and f(x)νj(x)h(x) = 0 for x ∈ Ω with νj(x) = 0. Hence
+fνjh is bounded on Ω. Further,
+εh(x) = ενi(x)/νj(x) ≥ 1,
+for x ∈ Ω ∖ K with νj(x) ≠ 0 because (20) is fulfilled. Moreover, the zeros of νj
+are contained in N ∶= {x ∈ Ω ∣ f(x)νj(x) = 0}. This yields that acx(fνj(Ω)) is
+absolutely convex and compact by Proposition A.1.4.
+So our statement follows
+from Corollary 3.2.5 (iv).
+□
+Let us apply the preceding result to some known sequence spaces. We recall
+that a matrix A ∶= (ak,j)k,j∈N of non-negative numbers is called Köthe matrix if it
+fulfils:
+(1) ∀ k ∈ N ∃ j ∈ N ∶ ak,j > 0,
+(2) ∀ k,j ∈ N ∶ ak,j ≤ ak,j+1.
+We note that what we call k is usually called j and vice-versa (see e.g. [131, Def-
+inition, p. 326]). But the notation we chose is more in line with the meaning of
+j in our Definition 3.1.3 of a weight function and therefore we prefer to keep our
+notation consistent. For an lcHs E we define the Köthe space
+λ∞(A,E) ∶= {x = (xk) ∈ EN ∣ ∀ j ∈ N, α ∈ A ∶ ∣x∣j,α ∶= sup
+k∈N
+pα(xk)ak,j < ∞}
+
+4.2. FURTHER EXAMPLES OF ε-PRODUCTS
+43
+and the spaces of E-valued rapidly decreasing sequences which we need for some
+theorems on Fourier expansions (see Theorem 5.6.13, Theorem 5.6.14) by
+s(Ω,E) ∶= {x = (xk) ∈ EΩ ∣ ∀ j ∈ N, α ∈ A ∶ ∣x∣j,α ∶= sup
+k∈Ω
+pα(xk)(1 + ∣k∣2)j/2 < ∞}
+with Ω = Nd, Nd
+0, Zd. Further, we set λ∞(A) ∶= λ∞(A,K) and s(Ω) ∶= s(Ω,K).
+4.2.3. Corollary. Let E be a locally complete lcHs.
+a) If A ∶= (ak,j)k,j∈N is a Köthe matrix such that
+∀ j ∈ N ∃ i ∈ N ∀ ε > 0 ∃ K ∈ N ∀ k ∈ N, k > K ∶ ak,j ≤ εak,i,
+(21)
+then λ∞(A,E) ≅ λ∞(A)εE.
+b) s(Ω,E) ≅ s(Ω)εE for Ω = Nd, Nd
+0, Zd.
+Proof. We observe that N and Ω are uniformly discrete metric spaces if they
+are equipped with the metric induced by the absolute value. Further, a set in a
+discrete space is compact if and only if it is finite. In case b) we set νj∶Ω → (0,∞),
+νj(k) ∶= (1 + ∣k∣2)j/2 for j ∈ N. Then for ε > 0 there is K ∈ N such that
+(1 + ∣k∣2)j/2
+(1 + ∣k∣2)j
+= (1 + ∣k∣2)−j/2 ≤ ε
+for all k ∈ Ω with ∣k∣ > K. In both cases the family of weights are directed, in case
+a) due to condition (2) of the definition of a Köthe matrix. Hence we can apply
+Example 4.2.2 in both cases.
+□
+Due to [131, Proposition 27.10, p. 330–331] condition (21) is equivalent to
+λ∞(A) being a Schwartz space.
+Since λ∞(A) is also a Fréchet space by [131,
+Lemma 27.1, p. 326], another way to prove Corollary 4.2.3 a) (and b) as well) is
+given by Corollary 3.2.5 (ii).
+Our next examples are Favard-spaces. Let E be an lcHs, 0 < γ ≤ 1, Ω a compact
+Hausdorff space, ϕ∶[0,∞) × Ω → Ω a continuous semiflow, i.e.
+ϕ(t + r,s) = ϕ(t,ϕ(r,s))
+and
+ϕ(0,s) = s,
+t,r ∈ [0,∞), s ∈ Ω,
+and (̃T E
+t )t≥0 the induced semigroup given by ̃T E
+t ∶C(Ω,E) → C(Ω,E), ̃T E
+t (f) ∶=
+f(ϕ(t,⋅)). The semigroup (̃T K
+t )t≥0 is (equi-)bounded and strongly continuous by
+[62, Chap. II, 3.31 Exercises (1), p. 95]. The vector-valued Favard space of order γ
+of the semigroup (̃T E
+t )t≥0 is defined by
+Fγ(Ω,E) ∶= {f ∈ C(Ω,E) ∣ ∀ α ∈ A ∶
+sup
+x∈Ω,t>0
+pα(̃T E
+t (f)(x) − f(x))t−γ < ∞}
+equipped with the system of seminorms given by
+∣f∣α ∶= max(sup
+x∈Ω
+pα(f(x)), sup
+x∈Ω,t>0
+pα(̃T E
+t (f)(x) − f(x))t−γ),
+f ∈ Fγ(Ω,E),
+for α ∈ A (see [39, Definition 3.1.2, p. 160] and [39, Proposition 3.1.3, p. 160]).
+Further, we set Fγ(Ω) ∶= Fγ(Ω,K). Fγ(Ω,E) is a dom-space, which follows from
+the setting ω ∶= [0,∞) × Ω, domT E ∶= C(Ω,E) and T E∶C(Ω,E) → Eω given by
+T E(f)(0,x) ∶= f(x) and T E(f)(t,x) ∶= ̃T E
+t (f)(x) − f(x),
+t > 0, x ∈ Ω,
+as well as AP(Ω,E) ∶= EΩ and the weight given by ν(0,x) ∶= 1 and ν(t,x) ∶= t−γ
+for t > 0 and x ∈ Ω.
+4.2.4. Example. Let E be a semi-Montel space, 0 < γ ≤ 1, Ω a compact Haus-
+dorff space, ϕ∶[0,∞) × Ω → Ω a continuous semiflow. Then Fγ(Ω,E) ≅ Fγ(Ω)εE
+holds for the Favard space of order γ of the induced semigroup (̃T E
+t )t≥0.
+
+44
+4. CONSISTENCY
+Proof. The generator (T E,T K) for (Fγ,E) is consistent by Proposition 4.1.1
+and Lemma 4.1.2 b)(ii). Its strength is clear. Thus our statement follows from
+Corollary 3.2.5 (iii).
+□
+The space of càdlàg functions on a set Ω ⊂ R with values in an lcHs E is defined
+by
+D(Ω,E) ∶= {f ∈ EΩ ∣ ∀ x ∈ Ω ∶
+lim
+w→x+f(w) = f(x) and lim
+w→x−f(w) exists}.2
+Further, we set D(Ω) ∶= D(Ω,K). Due to Proposition A.1.1 the maps given by
+∣f∣K,α ∶= sup
+x∈Ω
+pα(f(x))χK(x),
+f ∈ D(Ω,E),
+for compact K ⊂ Ω and α ∈ A form a system of seminorms inducing a locally convex
+Hausdorff topology on D(Ω,E).
+4.2.5. Example. Let E be an lcHs and Ω ⊂ R locally compact. If E is quasi-
+complete, then D(Ω)εE ≅ D(Ω,E).
+Proof. First, we show that the generator (idEΩ,idKΩ) for (D,E) is strong
+and consistent. The strength is a consequence of a simple calculation, so we only
+prove the consistency explicitly.
+We have to show that S(u) ∈ D(Ω,E) for all
+u ∈ D(Ω)εE. Let x ∈ Ω be an accumulation point of [x,∞) ∩ Ω resp. (−∞,x] ∩ Ω,
+(xn) be a sequence in Ω such that xn → x+ resp. xn → x−. We have
+δxn(f) = f(xn) → f(x) = δx(f),
+xn → x+,
+and
+δxn(f) = f(xn) → lim
+n→∞f(xn) =∶ T(f)(x),
+xn → x−,
+for every f ∈ D(Ω), which implies that (δxn) converges to δx if xn → x+, and to
+δx ○ T if xn → x− in D(Ω)′
+σ.
+Since Ω is locally compact, there are a compact
+neighbourhood U(x) ⊂ Ω of x and n0 ∈ N such that xn ∈ U(x) for all n ≥ n0. Hence
+we deduce
+sup
+n≥n0
+∣δxn(f)∣ ≤ ∣f∣U(x)
+for every f ∈ D(Ω). Therefore the set {δxn ∣ n ≥ n0} is equicontinuous in D(Ω)′,
+which implies that (δxn) converges to δx if xn → x+ and to δx ○ T if xn → x− in
+D(Ω)′
+γ and thus in D(Ω)′
+κ. From
+S(u)(x) = u(δx) = lim
+n→∞u(δxn) = lim
+n→∞S(u)(xn),
+xn → x+,
+and
+u(δx ○ T) = lim
+n→∞u(δxn) = lim
+n→∞S(u)(xn),
+xn → x−,
+for every u ∈ D(Ω)εE follows the consistency. Second, let f ∈ D(Ω,E), K ⊂ Ω be
+compact and consider NK(f) = f(Ω)χK(Ω). We observe that NK(f) = f(K)∪{0}
+if K ≠ Ω, and NK(f) = f(K) if K = Ω. We note that NK(f) ⊂ acx(f(K)) and
+acx(f(K)) is absolutely convex and compact by Proposition A.1.1 because E is
+quasi-complete. Thus we derive our statement from Corollary 3.2.5 (iv).
+□
+We turn to Cauchy continuous functions. Let Ω be a metric space, E an lcHs
+and the space CC(Ω,E) of Cauchy continuous functions from Ω to E be equipped
+with the system of seminorms given by
+∣f∣K,α ∶= sup
+x∈K
+pα(f(x))χK(x),
+f ∈ CC(Ω,E),
+for K ⊂ Ω precompact and α ∈ A.
+2We note that for x ∈ Ω we only demand limw→x+ f(w) = f(x) if x is an accumulation point of
+[x, ∞)∩Ω, and the existence of the limit limw→x− f(w) if x is an accumulation point of (−∞, x]∩Ω.
+
+4.2. FURTHER EXAMPLES OF ε-PRODUCTS
+45
+4.2.6. Example. Let E be an lcHs and Ω a metric space. If E is a Fréchet
+space or a semi-Montel space, then CC(Ω,E) ≅ CC(Ω)εE.
+Proof. The generator (idEΩ,idKΩ) for (CC,E) is consistent by Proposition
+4.1.3 with Lemma 4.1.4. Its strength follows from the uniform continuity of every
+e′ ∈ E′. First, we consider the case that E is a Fréchet space. Let f ∈ CC(Ω,E), K ⊂
+Ω be precompact and consider NK(f) = f(Ω)χK(Ω). Then NK(f) = f(K) ∪ {0} if
+K ≠ Ω, and NK(f) = f(K) if K = Ω. The set f(K) is precompact in the metrisable
+space E by [13, Proposition 4.11, p. 576]. Thus we obtain CC(Ω,E) ⊂ CC(Ω,E)κ
+by virtue of Lemma 3.2.2 c). Since E is complete, the first part of the statement
+follows from Theorem 3.2.4 with Condition 3.2.3 a). If E is a semi-Montel space,
+then it is a consequence of Corollary 3.2.5 (iii).
+□
+Let (Ω,d) be a metric space, E an lcHs and the space Cbu(Ω,E) of bounded
+uniformly continuous functions from Ω to E be equipped with the system of semi-
+norms given by
+∣f∣α ∶= sup
+x∈Ω
+pα(f(x)),
+f ∈ Cbu(Ω,E),
+for α ∈ A.
+4.2.7. Example. Let E be an lcHs and (Ω,d) a metric space. If E is a semi-
+Montel space, then Cbu(Ω,E) ≅ Cbu(Ω)εE.
+Proof. The generator (idEΩ,idKΩ) for (Cbu,E) is consistent by Proposition
+4.1.5 with Lemma 4.1.6. It is also strong due to the uniform continuity of every
+e′ ∈ E′, yielding our statement by Corollary 3.2.5 (iii).
+□
+4.2.8. Remark. If N is equipped with the metric induced by the absolut value,
+then Cbu(N,E) = ℓ∞(N,E) where ℓ∞(N,E) is the space of bounded E-valued
+sequences. If E is a separable infinite-dimensional Hilbert space, then the map
+S∶Cbu(N)εE → Cbu(N,E) is not surjective by [17, 2.8 Beispiel, p. 140] and [94, Satz
+10.5, p. 235–236]. Hence one cannot drop the condition that E is a semi-Montel
+space in Example 4.2.7.
+Let (Ω,d) be a metric space, z ∈ Ω, E an lcHs, 0 < γ ≤ 1 and define the space
+of E-valued γ-Hölder continuous functions on Ω that vanish at z by
+C[γ]
+z (Ω,E) ∶= {f ∈ EΩ ∣ f(z) = 0 and ∀ α ∈ A ∶ ∣f∣α < ∞}
+where
+∣f∣α ∶= sup
+x,w∈Ω
+x≠w
+pα(f(x) − f(w))
+d(x,w)γ
+.
+The topological subspace C[γ]
+z,0(Ω,E) of γ-Hölder continuous functions that vanish
+at infinity consists of all f ∈ C[γ]
+z (Ω,E) such that for all ε > 0 there is δ > 0 with
+sup
+x,w∈Ω
+0<d(x,w)<δ
+pα(f(x) − f(w))
+d(x,w)γ
+< ε.
+Further, we set C[γ]
+z (Ω) ∶= C[γ]
+z (Ω,K) and C[γ]
+z,0(Ω) ∶= C[γ]
+z,0(Ω,K).
+Moreover, we
+define M ∶= J ∶= {1}, ω1 ∶= Ω2 ∖ {(x,x) ∣ x ∈ Ω} and T E
+1 ∶EΩ → Eω1, T E
+1 (f)(x,w) ∶=
+f(x) − f(w), and
+ν1,1∶ω1 → [0,∞), ν1,1(x,w) ∶=
+1
+d(x,w)γ .
+Then we have for every α ∈ A that
+∣f∣α =
+sup
+(x,w)∈ω1
+pα(T E
+1 (f)(x,w))ν1,1(x,w),
+f ∈ C[γ]
+z (Ω,E).
+
+46
+4. CONSISTENCY
+4.2.9. Example. Let E be an lcHs, (Ω,d) a metric space, z ∈ Ω and 0 < γ ≤ 1.
+Then
+a) C[γ]
+z (Ω,E) ≅ C[γ]
+z (Ω)εE if E is a semi-Montel space,
+b) C[γ]
+z,0(Ω,E) ≅ C[γ]
+z,0(Ω)εE if Ω is precompact and E quasi-complete.
+Proof. Let us start with a). From Proposition 4.1.10 for vanishing at z and
+a simple calculation follows that (T E
+1 ,T K
+1 ) is a strong and consistent generator for
+(C[γ]
+z ,E). This proves part a) by Corollary 3.2.5 (iii). Concerning part b), we set
+K ∶= {{(x,w) ∈ Ω2 �� d(x,w) ≥ δ} ∣ δ > 0}, and let π∶ω1 → ω1 be the identity. Then
+C[γ]
+z,0(Ω,E) = C[γ]
+z (Ω,E) ∩ APπ,K(Ω,E) with APπ,K(Ω,E) from Proposition 4.1.9
+and the generator (T E
+1 ,T K
+1 ) for (C[γ]
+z,0,E) is strong and consistent by Proposition
+4.1.10 for vanishing at z and Proposition 4.1.9 for vanishing at infinity w.r.t. (π,K).
+Let f ∈ C[γ]
+z,0(Ω,E) and Kδ ∶= {(x,w) ∈ Ω2 ∣ d(x,w) ≥ δ} for δ > 0. For
+Nπ⊂Kδ,1,1(f) = {T E
+1 (f)(x,w)ν1,1(x,w) ∣ (x,w) ∈ Kδ} = { f(x)−f(w)
+d(x,w)γ
+∣ (x,w) ∈ Kδ}
+we have
+Nπ⊂Kδ,1,1(f) ⊂ δ−γ{c(f(x) − f(w)) ∣ x,w ∈ Ω, ∣c∣ ≤ 1}
+= δ−γ ch(f(Ω) − f(Ω)).
+The set f(Ω) is precompact because Ω is precompact and the γ-Hölder continuous
+function f is uniformly continuous. It follows that the linear combination f(Ω) −
+f(Ω) is precompact and the circled hull of a precompact set is still precompact by
+[153, Chap. I, 5.1, p. 25]. Therefore Nπ⊂Kδ,1,1(f) is precompact for every δ > 0,
+giving the precompactness of
+N1,1(f) = {T E
+1 (f)(x,w)ν1,1(x,w) ∣ (x,w) ∈ ω1}
+by Proposition A.1.6. Hence statement b) is a consequence of Corollary 3.2.5 (iv),
+Proposition A.1.6 and the quasi-completeness of E.
+□
+Let Ω be a topological Hausdorff space and V ∶= (νj)j∈J a directed family of
+weights νj∶Ω → [0,∞).
+The weighted space of continuous functions on Ω with
+values in an lcHs E is given by
+CV(Ω,E) ∶= {f ∈ C(Ω,E) ∣ ∀ j ∈ J, α ∈ A ∶ ∣f∣j,α < ∞}
+where
+∣f∣j,α ∶= sup
+x∈Ω
+pα(f(x))νj(x).
+Its topological subspace of functions that vanish at infinity in the weighted topology
+is defined by
+CV0(Ω,E) ∶= {f ∈ CV(Ω,E) ∣ ∀ j ∈ J, α ∈ A, ε > 0
+∃ K ⊂ Ω compact ∶ ∣f∣Ω∖K,j,α < ε}
+where
+∣f∣Ω∖K,j,α ∶= sup
+x∈Ω∖K
+pα(f(x))νj(x).
+Further, we define CV(Ω) ∶= CV(Ω,K) and CV0(Ω) ∶= CV(Ω,K). In particular, we
+set Cb(Ω,E) ∶= CV(Ω,E), i.e. the space of bounded continuous functions, and have
+CV0(Ω,E) = C0(Ω,E) if V ∶= {1}. In [15, 16, 17] Bierstedt studies these spaces in the
+case that V is a Nachbin-family which means that the functions νj are upper semi-
+continuous for all j ∈ J and directed in the sense that for j1,j2 ∈ J and λ ≥ 0 there
+is j3 ∈ J such that λνj1,λνj2 ≤ νj3. Formally this is stronger than our definition
+of being directed in Remark 3.1.6 c). The notion U ≤ V for two Nachbin-families
+
+4.2. FURTHER EXAMPLES OF ε-PRODUCTS
+47
+means that for every µ ∈ U there is ν ∈ V such that µ ≤ ν. One of his main results
+from [17] is the following theorem.
+4.2.10. Theorem ([17, 2.4 Theorem (2), p. 138–139]). Let E be a quasi-
+complete lcHs, Ω a completely regular Hausdorff space and V a Nachbin-family
+on Ω. If
+(i) Z ∶= {v∶Ω → R ∣ v constant, v ≥ 0} ≤ V, or
+(ii) ̃
+W ∶= {µχK ∣ µ > 0, K ⊂ Ω compact} ≤ V and Ω is a kR-space,
+then CV0(Ω,E) ≅ CV0(Ω)εE.
+We note that C̃
+W(Ω,E) = CW(Ω,E) with our definition of W = {χK ∣ K ⊂
+Ω compact} from above Lemma 4.1.2.
+The only difference is that W is not a
+Nachbin-family because it is not directed in the sense of Nachbin-families but in
+the sense of Remark 3.1.6 c). We improve this result by strengthening the conditions
+on Ω and V which allows us to weaken the assumptions on E.
+4.2.11. Example. Let E be an lcHs, Ω a locally compact topological Hausdorff
+space and V a directed family of continuous weights on Ω.
+(i) If E has ccp, or
+(ii) if E has metric ccp and Ω is second-countable,
+then CV0(Ω,E) ≅ CV0(Ω)εE.
+Proof. We set K ∶= {K ⊂ Ω ∣ K compact} and π∶Ω → Ω, π(x) ∶= x. It follows
+from Proposition 4.1.1 combined with Lemma 4.1.2 (i) (continuity) and Proposition
+4.1.9 (vanish at infinity w.r.t. (π,K)) that the generator (idEΩ,idKΩ) is strong and
+consistent since V is a family of continuous weights and Ω a kR-space due to local
+compactness.
+Let f ∈ CV0(Ω,E), j ∈ J and consider Nj(f) = (fνj)(Ω).
+By Proposition
+A.1.3 the set K ∶= acx(Nj(f)) is absolutely convex and compact as fνj ∈ C0(Ω,E),
+implying our statement by Corollary 3.2.5 (iv).
+□
+4.2.12. Example. Let E be an lcHs and Ω a [metrisable] kR-space. If E has
+[metric] ccp, then CW(Ω,E) ≅ CW(Ω)εE.
+Proof. First, we observe that the generator (idEΩ,idKΩ) for (CW,E) is con-
+sistent by Proposition 4.1.1 and Lemma 4.1.2 b)(i). Its strength is obvious. Let
+f ∈ CW(Ω,E), K ⊂ Ω be compact and consider NK(f) = f(Ω)νK(Ω).
+Then
+NK(f) = f(K) ∪ {0} if K ≠ Ω, and NK(f) = f(K) if K = Ω, which yields that
+NK(f) is compact in E. If Ω is even metrisable, then f(K) is also metrisable by
+[34, Chap. IX, §2.10, Proposition 17, p. 159] and thus the finite union NK(f) as
+well by [169, Theorem 1, p. 361] since the compact set NK(f) is collectionwise
+normal and locally countably compact by [63, 5.1.18 Theorem, p. 305]. Further,
+acx(NK(f)) is absolutely convex and compact in E if E has ccp resp. if Ω is metris-
+able and E has metric ccp. We conclude that CW(Ω,E) ≅ CW(Ω)εE if E has ccp
+resp. if Ω is metrisable and E has metric ccp by Corollary 3.2.5 (iv).
+□
+Bierstedt also considers closed subspaces of CV(Ω) and CV0(Ω), for instance
+subspaces of holomorpic functions on open Ω, and of holomorpic functions on the
+inner points of Ω which are continuous on the boundary in [17, 3.1 Bemerkung, p.
+141] and [17, 3.7 Satz, p. 144].
+Let Ω ⊂ C be open and bounded and E an lcHs over C. We denote by A(Ω,E)
+the space of continuous functions from Ω to an lcHs E which are holomorphic on
+Ω and equip A(Ω,E) with the system of seminorms given by
+∣f∣α ∶= sup
+x∈Ω
+pα(f(x)),
+f ∈ A(Ω,E),
+
+48
+4. CONSISTENCY
+for α ∈ A. We set A(Ω) ∶= A(Ω,C), J ∶= M ∶= {1} and ν1,1 ∶= 1 on Ω.
+4.2.13. Example. Let E be an lcHs and Ω ⊂ C open and bounded.
+Then
+A(Ω,E) ≅ A(Ω)εE if E has metric ccp.
+Proof. The space A(Ω) is a Banach space and hence barrelled. The inclusion
+I∶A(Ω) → CW∞
+∂ (Ω) is continuous due to the Cauchy inequality (I is an inclusion
+due to the identity theorem). It follows from Proposition 4.1.1, Lemma 4.1.2 b)(i),
+Proposition 3.1.11 c) and (4) that the generator (idEΩ,idCΩ) is consistent and as
+in Proposition 3.1.10 that it is strong, too.
+Let f ∈ A(Ω,E) and N1,1(f) = f(Ω). The set K ∶= acx(N1,1(f)) is absolutely
+convex and compact by Proposition A.1.3 since f ∈ C(Ω,E) = C0(Ω,E), implying
+our statement by Corollary 3.2.5 (iv).
+□
+For quasi-complete E this is already covered by [17, 3.1 Bemerkung, p. 141].
+More general than holomorphic functions, we may also consider kernels of hypoel-
+liptic linear partial differential operators in CV(Ω) and CV0(Ω). For an open set
+Ω ⊂ Rd, a directed family V ∶= (νj)j∈N of weights νj∶Ω → [0,∞), an lcHs E and a
+linear partial differential operator P(∂)E which is hypoelliptic if E = K we define
+the space of zero solutions
+CVP (∂)(Ω,E) ∶= {f ∈ C∞
+P (∂)(Ω,E) ∣ ∀ j ∈ N, α ∈ A ∶ ∣f∣j,α < ∞},
+where C∞
+P (∂)(Ω,E) is the kernel of P(∂)E in C∞(Ω,E),
+∣f∣j,α ∶= sup
+x∈Ω
+pα(f(x))νj(x),
+and its topological subspace
+CV0,P (∂)(Ω,E) ∶= CVP (∂)(Ω,E) ∩ CV0(Ω,E).
+Further, we set CVP (∂)(Ω) ∶= CVP (∂)(Ω,K) and CV0,P (∂)(Ω) ∶= CV0,P (∂)(Ω,K). We
+say that V is locally bounded away from zero on Ω if
+∀ K ⊂ Ω compact ∃ j ∈ N ∶ inf
+x∈K νj(x) > 0.
+This is an extension of the definition of being locally bounded away from zero from
+Vk with k ∈ N∞ to the case k = 0 (see Proposition 3.1.10). If V is a Nachbin-family,
+this means that ̃
+W ≤ V (see Theorem 4.2.10 (ii)).
+4.2.14. Proposition. Let Ω ⊂ Rd be open, V ∶= (νj)j∈N an increasing family
+of weights which is locally bounded away from zero on Ω and P(∂)K a hypoelliptic
+linear partial differential operator. Then CVP (∂)(Ω) and CV0,P (∂)(Ω) are Fréchet
+spaces.
+Proof. We note that CVP (∂)(Ω) is metrisable as V is countable. Let (fn) be
+a Cauchy sequence in CVP (∂)(Ω). From V being locally bounded away from zero
+it follows that for every compact K ⊂ Ω there is j ∈ N such that
+sup
+x∈K
+∣f(x)∣ ≤ sup
+z∈K
+νj(z)−1 sup
+x∈K
+∣f(x)∣νj(x) ≤ sup
+z∈K
+νj(z)−1∣f∣j,
+f ∈ CVP (∂)(Ω),
+(22)
+which means that the inclusion I∶CVP (∂)(Ω) → CWP (∂)(Ω) is continuous. Thus
+(fn) is also a Cauchy sequence in CWP (∂)(Ω) and has a limit f there as CWP (∂)(Ω)
+is complete due to the hypoellipticity of P(∂)K. Let j ∈ N, ε > 0 and x ∈ Ω. Then
+there is mj,ε,x ∈ N such that for all m ≥ mj,ε,x it holds that
+∣fm(x) − f(x)∣ <
+ε
+2νj(x)
+
+4.2. FURTHER EXAMPLES OF ε-PRODUCTS
+49
+if νj(x) ≠ 0. Further, there is mj,ε ∈ N such that for all n,m ≥ mj,ε it holds that
+∣fn − fm∣j < ε
+2.
+Hence for n ≥ mj,ε we choose m ≥ max(mj,ε,mj,ε,x) and derive
+∣fn(x)−f(x)∣νj(x) ≤ ∣fn(x)−fm(x)∣νj(x)+∣fm(x)−f(x)∣νj(x) < ε
+2+
+ε
+2νj(x)νj(x) = ε.
+It follows that ∣fn − f∣j ≤ ε and ∣f∣j ≤ ε + ∣fn∣j for all n ≥ mj,ε, implying the
+convergence of (fn) to f in CVP (∂)(Ω). Therefore CVP (∂)(Ω) is a Fréchet space.
+CV0,P (∂)(Ω) is a closed subspace of CVP (∂)(Ω) and so a Fréchet space as well.
+□
+Due to the proposition above the spaces CVP (∂)(Ω) and CV0,P (∂)(Ω) are closed
+subspaces of CV(Ω) resp. CV0(Ω). Hence we have the following consequence of
+Theorem 4.2.10 (ii), [17, 2.12 Satz (1), p. 141] and [17, 3.1 Bemerkung, p. 141].
+4.2.15. Corollary. Let E be an lcHs, Ω ⊂ Rd open, V a Nachbin-family on
+Ω which is locally bounded away from zero and P(∂)K a hypoelliptic linear partial
+differential operator.
+a) CVP (∂)(Ω,E) ≅ CVP (∂)(Ω)εE if E is a semi-Montel space.
+b) CV0,P (∂)(Ω,E) ≅ CV0,P (∂)(Ω)εE if E is quasi-complete.
+Like before we may improve this result by strengthening the conditions on V
+and CVP (∂)(Ω) resp. CV0,P (∂)(Ω) which allows us to weaken the assumptions on
+E.
+4.2.16. Example. Let E be an lcHs, Ω ⊂ Rd open, V ∶= (νj)j∈N an increasing
+family of weights which is locally bounded away from zero on Ω and P(∂)K a
+hypoelliptic linear partial differential operator.
+a) CVP (∂)(Ω,E) ≅ CVP (∂)(Ω)εE if E is complete and CVP (∂)(Ω) a semi-
+Montel space.
+b) CVP (∂)(Ω,E) ≅ CVP (∂)(Ω)εE if E is locally complete and CVP (∂)(Ω) a
+Schwartz space.
+c) CV0,P (∂)(Ω,E) ≅ CV0,P (∂)(Ω)εE if E has metric ccp and νj ∈ C(Ω) for all
+j ∈ N.
+d) CV0,P (∂)(Ω,E) ≅ CV0,P (∂)(Ω)εE if E is locally complete and CV0,P (∂)(Ω)
+a Schwartz space.
+Proof. Let F stand for CVP (∂) or CV0,P (∂). The space F(Ω) is a Fréchet space
+and hence barrelled by Proposition 4.2.14. The inclusion I∶F(Ω) → CWP (∂)(Ω) is
+continuous since V is locally bounded away from zero on Ω. The hypoellipticity
+of P(∂)K (see e.g. [70, p. 690]) yields that CWP (∂)(Ω) = CW∞
+P (∂)(Ω) as locally
+convex spaces. Thus the inclusion I∶F(Ω) → CW∞
+P (∂)(Ω) is continuous. It follows
+from Proposition 3.1.11 c) that the generator (idEΩ,idKΩ) is consistent if F =
+CVP (∂), and combined with Proposition 4.1.9 (vanish at infinity w.r.t. (π,K)) if
+F = CV0,P (∂) where K and π are chosen as in Example 4.2.11. The strength of the
+generator follows as in Proposition 3.1.10 and, if F = CV0,P (∂), in combination with
+Proposition 4.1.9 b). This proves part a), b) and d) due to Corollary 3.2.5 (i) and
+(ii).
+Let us turn to part c). Let f ∈ CV0,P (∂)(Ω,E), j ∈ N and Nj(f) ∶= (fνj)(Ω).
+The set K ∶= acx(Nj(f)) is absolutely convex compact by Proposition A.1.3 as
+fνj ∈ C0(Ω,E), implying our statement by Corollary 3.2.5 (iv).
+□
+At least for some weights and operators P(∂) we can show that CVP (∂)(Ω,E)
+coincides with a corresponding space CV∞
+P (∂)(Ω,E) from Example 3.1.9 if E is
+locally complete.
+
+50
+4. CONSISTENCY
+4.2.17. Proposition. Let E be a locally complete lcHs, Ω ⊂ Rd and P(∂)K
+a hypoelliptic linear partial differential operator. Then we have CWP (∂)(Ω)εE ≅
+CWP (∂)(Ω,E) and CWP (∂)(Ω,E) = CW∞
+P (∂)(Ω,E) as locally convex spaces.
+Proof. We already know that
+SCW∞
+P (∂)(Ω)∶CW∞
+P (∂)(Ω)εE → CW∞
+P (∂)(Ω,E)
+is an isomorphism by Example 3.2.8 b).
+The hypoellipticity of P(∂)K (see e.g.
+[70, p. 690]) yields that CWP (∂)(Ω)εE = CW∞
+P (∂)(Ω)εE. Thus SCWP (∂)(Ω)(u) =
+SCW∞
+P (∂)(Ω)(u) ∈ C∞
+P (∂)(Ω,E) for all u ∈ CWP (∂)(Ω)εE. In particular, we obtain
+that
+SCWP (∂)(Ω)∶CWP (∂)(Ω)εE → CW∞
+P (∂)(Ω,E)
+is an isomorphism. From Proposition 3.1.11 c) and Theorem 3.1.12 with (T E,T K) ∶=
+(idEΩ,idKΩ) we deduce that
+SCWP (∂)(Ω)∶CWP (∂)(Ω)εE → CWP (∂)(Ω,E)
+is an isomorphism into, and from
+SCWP (∂)(Ω)(CWP (∂)(Ω)εE) = C∞
+P (∂)(Ω,E)
+that CWP (∂)(Ω,E) = CW∞
+P (∂)(Ω,E) as locally convex spaces, which proves our
+statement.
+□
+Hence the topology τc of compact convergence induced by C(Ω,E) and the
+usual topology from Example 3.1.2 induced by C∞(Ω,E) coincide on CP (∂)(Ω,E) if
+P(∂)K is hypoelliptic and E locally complete by Proposition 4.2.17. In particular,
+we have
+(O(Ω,E),τc) =
+(4) CW∂(Ω,E) = CW∞
+∂ (Ω,E)
+(23)
+if E is locally complete. For more interesting weights than W we introduce the
+following condition.
+4.2.18. Condition. Let V ∶= (νj)j∈N be an increasing family of continuous
+weights on Rd. Let there be r∶Rd → (0,1] and for any j ∈ N let there be ψj ∈ L1(Rd),
+ψj > 0, and N ∋ Im(j) ≥ j and Am(j) > 0, m ∈ {1,2,3}, such that for any x ∈ Rd:
+(α.1) supζ∈Rd, ∥ζ∥∞≤r(x) νj(x + ζ) ≤ A1(j)infζ∈Rd, ∥ζ∥∞≤r(x) νI1(j)(x + ζ),
+(α.2) νj(x) ≤ A2(j)ψj(x)νI2(j)(x),
+(α.3) νj(x) ≤ A3(j)r(x)νI3(j)(x).
+Here, ∥ζ∥∞ ∶= sup1≤n≤d ∣ζn∣ for ζ = (ζn) ∈ Rd. The preceding condition is a
+special case of [111, Condition 2.1, p. 176] with Ω ∶= Ωn ∶= Rd for all n ∈ N. If V
+fulfils Condition 4.2.18 and we set V∞ ∶= (νj,m)j∈N,m∈N0 where νj,m∶{β ∈ Nd
+0 ∣ ∣β∣ ≤
+m} × Rd → [0,∞), νj,m(β,x) ∶= νj(x), then CV∞(Rd) and its closed subspace
+CV∞
+P (∂)(Rd) for P(∂) with continuous coefficients are nuclear by [111, Theorem
+3.1, p. 188] in combination with [111, Remark 2.7, p. 178–179] and Fréchet spaces
+by [107, Proposition 3.7, p. 240].
+4.2.19. Proposition. Let E be a locally complete lcHs, V ∶= (νj)j∈N an in-
+creasing family of continuous weights on Rd and V∞ defined as above. If V ful-
+fils Condition 4.2.18, then CV∂(C) and CV∆(Rd) are nuclear Fréchet spaces and
+CV∂(C,E) = CV∞
+∂ (C,E) and CV∆(Rd,E) = CV∞
+∆(Rd,E) as locally convex spaces.
+Proof. Let P(∂) ∶= ∂ (d ∶= 2 and K ∶= C) or P(∂) ∶= ∆. First, we show that
+CVP (∂)(Rd) = CV∞
+P (∂)(Rd) as locally convex spaces, which implies that CVP (∂)(Rd)
+is a nuclear Fréchet space as CV∞
+P (∂)(Rd) is such a space. Let f ∈ CV∂(C), j ∈ N,
+
+4.2. FURTHER EXAMPLES OF ε-PRODUCTS
+51
+m ∈ N0, z ∈ C and β ∶= (β1,β2) ∈ N2
+0. Then it follows from ∥ ⋅ ∥∞ ≤ ∣ ⋅ ∣ and Cauchy’s
+inequality that
+∣∂βf(z)∣νj(z) =
+(5) ∣iβ2∂∣β∣
+C f(z)∣νj(z) ≤
+∣β∣!
+r(z)∣β∣
+sup
+∣w−z∣=r(z)
+∣f(w)∣νj(z)
+≤
+(α.3)∣β∣!C(j,∣β∣)
+sup
+∣w−z∣=r(z)
+∣f(w)∣νB3(j)(z)
+≤
+(α.1)∣β∣!C(j,∣β∣)A1(B3(j))
+sup
+∣w−z∣=r(z)
+∣f(w)∣νI1B3(j)(w)
+≤ ∣β∣!C(j,∣β∣)A1(B3(j))∣f∣CV∂(C),I1B3(j)
+where C(j,∣β∣) ∶= A3(j)A3(I3(j))⋯A3((B3 − 1)(j)) and B3 − 1 is the (∣β∣ − 1)-fold
+composition of I3. Choosing k ∶= max∣β∣≤m I1B3(j), it follows that
+∣f∣CV∞
+∂ (C),j,m ≤ sup
+∣β∣≤m
+∣β∣!C(j,∣β∣)A1(B3(j))∣f∣CV∂(C),k < ∞
+and thus f ∈ CV∞
+∂ (C) and CV∂(C) = CV∞
+∂ (C) as locally convex spaces.
+In the
+case P(∂) = ∆ an analogous proof works due to Cauchy’s inequality for harmonic
+functions, i.e. for all f ∈ CV∆(Rd), j ∈ N, x ∈ Rd and β ∈ Nd
+0 it holds that
+∣∂βf(x)∣νj(x) ≤ ( d∣β∣
+r(x))
+∣β∣
+sup
+∣w−x∣<r(x)
+∣f(w)∣νj(x)
+(see e.g. [74, Theorem 2.10, p. 23]).
+Nuclear Fréchet spaces are Fréchet–Schwartz spaces and hence we have
+CV∞
+P (∂)(Rd,E) ≅ CV∞
+P (∂)(Rd)εE ≅ CVP (∂)(Rd)εE ≅ CVP (∂)(Rd,E)
+by Example 3.2.7 f) and Example 4.2.16 b). The isomorphism CV∞
+P (∂)(Rd,E) ≅
+CVP (∂)(Rd,E) is
+S−1
+CV∞
+P (∂)(Rd) ○ idCV∞
+P (∂)(Rd)εE ○SCVP (∂)(Rd) = S−1
+CV∞
+P (∂)(Rd) ○ SCVP (∂)(Rd) = idCV∞
+P (∂)(Rd,E)
+by Theorem 3.2.4.
+□
+4.2.20. Remark. Let E be a locally complete lcHs and P(∂)K a hypoelliptic
+linear partial differential operator.
+a) Let 0 ≤ τ < ∞. Then V ∶= (νj)j∈N given by νj(x) ∶= exp(−(τ + 1
+j )∣x∣),
+x ∈ Rd, fulfils Condition 4.2.18 by [111, Example 2.8 (iii), p. 179]. Then
+CVP (∂)(Rd,E) is the space of smooth functions of exponential type τ in
+the kernel of P(∂). If τ = 0, then the elements of these spaces are also
+called functions of infra-exponential type. In particular, if P(∂) = ∂, d = 2
+and K = C, or P(∂) = ∆, then Aτ
+∂(C,E) ∶= CV∂(C,E) is the space of
+entire and Aτ
+∆(Rd,E) ∶= CV∆(Rd,E) the space of harmonic functions of
+exponential type τ.
+b) Further examples of families of weights fulfilling Condition 4.2.18 can be
+found in [111, Example 2.8, p. 179] and [130, 1.5 Examples, p. 205].
+Next, we take a look at k-times continuously partially differentiable functions
+that vanish with all their derivatives when weighted at infinity. Let k ∈ N∞, Ω ⊂ Rd
+be open, ωm ∶= Mm × Ω with Mm ∶= {β ∈ Nd
+0 ∣ ∣β∣ ≤ min(m,k)} for all m ∈ N0
+and Vk ∶= (νj,m)j∈J,m∈N0 be a directed family of weights on (ωm)m∈N0. We define
+the topological subspace of CVk(Ω,E) from Example 3.1.9 a)(i) consisting of the
+functions that vanishs with all their derivatives when weighted at infinity by
+CVk
+0(Ω,E) ∶= {f ∈ CVk(Ω,E) ∣ ∀ j ∈ J, m ∈ N0, α ∈ A, ε > 0
+∃ K ⊂ Ω compact ∶ ∣f∣Ω∖K,j,m,α < ε}
+
+52
+4. CONSISTENCY
+where
+∣f∣Ω∖K,j,m,α ∶= sup
+x∈Ω∖K
+β∈Mm
+pα((∂β)Ef(x))νj,m(β,x).
+Further, we define its subspace CVk
+0,P (∂)(Ω,E) ∶= {f ∈ CVk
+0(Ω,E) ∣ f ∈ kerP(∂)E}
+where
+P(∂)E∶Ck(Ω,E) → EΩ, P(∂)E(f)(x) ∶=
+n
+∑
+i=1
+ai(x)(∂βi)E(f)(x),
+with n ∈ N, βi ∈ Nd
+0 such that ∣βi∣ ≤ k and ai∶Ω → K for 1 ≤ i ≤ n.
+4.2.21. Remark. If Vk fulfils condition (V∞) from Example 3.2.9, then we have
+CVk
+0(Ω,E) = CVk(Ω,E) (see [107, Remark 3.4, p. 239]).
+So CWk(Ω,E), S(Rd,E) and OM(Rd,E) are concrete examples of spaces
+CVk
+0(Ω,E) (see Corollary 3.2.10).
+We present the counterpart for differentiable
+functions to Bierstedt’s Theorem 4.2.10 for the space CV0(Ω,E) of continuous func-
+tions from a completely regular Hausdorff space Ω to an lcHs E weighted with a
+Nachbin-family V that vanish at infinity in the weighted topology. For this purpose
+we need the following definition. We call Vk locally bounded on Ω if
+∀ K ⊂ Ω compact, j ∈ J, m ∈ N0, β ∈ Mm ∶ sup
+x∈K
+νj,m(β,x) < ∞.
+4.2.22. Example. Let E be an lcHs, k ∈ N∞, Vk be a directed family of weights
+which is locally bounded away from zero on an open set Ω ⊂ Rd.
+a) CVk
+0(Ω,E) ≅ CVk
+0(Ω)εE if E is quasi-complete, Vk locally bounded and
+CVk
+0(Ω) barrelled.
+b) CVk
+0(Ω,E) ≅ CVk
+0(Ω)εE if E has metric ccp, CVk
+0(Ω) is barrelled and
+νj,m(β,⋅) ∈ C(Ω) for all j ∈ J, m ∈ N0, β ∈ Nd
+0, ∣β∣ ≤ min(m,k).
+c) CVk
+0(Ω,E) ≅ CVk
+0(Ω)εE if E is locally complete and CVk
+0(Ω) a Fréchet–
+Schwartz space.
+d) CVk
+0,P (∂)(Ω,E) ≅ CVk
+0,P (∂)(Ω)εE if E is quasi-complete, Vk loc. bounded
+and CVk
+0,P (∂)(Ω) barrelled.
+e) CVk
+0,P (∂)(Ω,E) ≅ CVk
+0,P (∂)(Ω)εE if E has metric ccp, CVk
+0,P (∂)(Ω) is bar-
+relled and νj,m(β,⋅) ∈ C(Ω) for all j ∈ J, m ∈ N0, β ∈ Nd
+0, ∣β∣ ≤ min(m,k).
+f) CVk
+0,P (∂)(Ω,E) ≅ CVk
+0,P (∂)(Ω)εE if E is locally complete and CVk
+0,P (∂)(Ω)
+a Fréchet–Schwartz space.
+Proof. The generator (T E
+m,T K
+m)m∈N0 for (CVk
+0,E) and (CVk
+0,P (∂),E) is given
+by domT E
+m ∶= Ck(Ω,E) and
+T E
+m∶Ck(Ω,E) → Eωm, f �→ [(β,x) ↦ (∂β)Ef(x)],
+for all m ∈ N0 and the same with K instead of E.
+Set X ∶= Ω, K ∶= {K ⊂ Ω ∣ K compact} and π∶⋃m∈N0 ωm → X, π(β,x) ∶= x. We
+have
+∣f∣Ω∖K,j,m,α =
+sup
+x∈ωm
+π(x)∉K
+pα(T E
+m(f)(x))νj,m(x),
+for f ∈ CVk
+0(Ω,E), K ∈ K, j ∈ J and m ∈ N0, implying that (16) is satisfied. With
+APπ,K(Ω,E) from Proposition 4.1.9 we note that
+CVk
+0(Ω,E) = CVk(Ω,E) ∩ APπ,K(Ω,E).
+As in Proposition 3.1.10 it follows that the generator (T E
+m,T K
+m)m∈N0 fulfils (17) and
+(18) where we use Proposition 3.1.11, the barrelledness of CVk
+0(Ω) resp. CVk
+0,P (∂)(Ω)
+
+4.2. FURTHER EXAMPLES OF ε-PRODUCTS
+53
+and the assumption that Vk is locally bounded away from zero on Ω. Therefore the
+generator is strong and consistent by virtue of Proposition 4.1.9.
+a)+d) Let f ∈ CVk
+0(Ω,E), K ∈ K, j ∈ J and m ∈ N0. We claim that the set
+Nj,m(f) = {∂β)Ef(x)νj,m(β,x) ∣ x ∈ Ω, β ∈ Mm}
+is precompact in E by Proposition A.1.6. Since f vanishes at infinity in the weighted
+topology, condition (i) of Proposition A.1.6 is fulfilled. Hence we only need to show
+that condition (ii) is satisfied as well, i.e. we have to show that
+Nπ⊂K,j,m(f) =
+⋃
+β∈Mm
+(∂β)Efνj,m(β,⋅)(K)
+is precompact in E. Thus we only have to prove that the sets (∂β)Efνj,m(β,⋅)(K)
+are precompact since Nπ⊂K,j,m(f) is a finite union of these sets.
+But this is a
+consequence of the proof of [15, §1, 16. Lemma, p. 15] using the continuity of
+(∂β)Ef and the boundedness of νj,m(β,K), which follows from Vk being locally
+bounded. So we deduce statements a) and d) from Corollary 3.2.5 (iv), Proposition
+A.1.6 and the quasi-completeness of E.
+b)+e) The set Kβ ∶= acx((∂β)Efνj,m(β,⋅)(Ω)) is absolutely convex and com-
+pact by Proposition A.1.3 (ii) for every f ∈ CVk
+0(Ω,E), j ∈ J, m ∈ N0 and β ∈ Mm
+as E has metric ccp and νj,m(β,⋅) ∈ C(Ω). We have
+Nj,m(f) = {(∂β)Ef(x)νj,m(β,x) ∣ x ∈ Ω, β ∈ Mm} ⊂ acx( ⋃
+β∈Mm
+Kβ)
+and the set on the right-hand side is absolutely convex and compact by [89, 6.7.3
+Proposition, p. 113]. Now, statements b)+e) follow from Corollary 3.2.5 (iv).
+c)+f) They follow from Corollary 3.2.5 (ii).
+□
+The spaces CVk
+0(Ω) are Fréchet spaces and thus barrelled if J is countable by
+[107, Proposition 3.7, p. 240]. In [107, Theorem 5.2, p. 255] the question is answered
+when they have the approximation property. The spaces CV∞
+0 (Ω) and CV∞
+P (∂),0(Ω)
+are closed subspaces of CV∞(Ω) and CV∞
+P (∂)(Ω), respectively. For conditions that
+they are Fréchet–Schwartz spaces see the remarks below Example 3.2.7.
+We already saw different choices for K in Example 4.2.9 b) and Example 4.2.22.
+For holomorphic functions on an open subset Ω of an infinite dimensional Banach
+space X the family K of Ω-bounded sets, i.e. bounded sets K ⊂ Ω with positive
+distance to X ∖ Ω, is used in [71, p. 2] and [93, p. 2]. This family is clearly closed
+under taking finite unions, so Proposition 4.1.9 is applicable as well.
+Now, we consider an example of weighted smooth functions where the corre-
+sponding space of scalar-valued functions may not be barrelled. For an open set
+Ω ⊂ Rd, an lcHs E and a linear partial differential operator P(∂)E which is hypoel-
+liptic if E = K we define the space of bounded zero solutions
+C∞
+P (∂),b(Ω,E) ∶= {f ∈ C∞
+P (∂)(Ω,E) ∣ ∀ α ∈ A ∶ ∥f∥∞,α ∶= sup
+x∈Ω
+pα(f(x)) < ∞}
+where C∞
+P (∂)(Ω,E) is the kernel of P(∂)E in C∞(Ω,E). Further, we set C∞
+P (∂),b(Ω) ∶=
+C∞
+P (∂),b(Ω,K).
+Apart from the topology given by (∥ ⋅ ∥∞,α)α∈A there is another
+weighted locally convex topology on C∞
+P (∂),b(Ω,E) which is of interest, namely, the
+one induced by the seminorms
+∣f∣ν,α ∶= sup
+x∈Ω
+pα(f(x))∣ν(x)∣,
+f ∈ C∞
+P (∂),b(Ω,E),
+for ν ∈ C0(Ω) and α ∈ A. We denote by (C∞
+P (∂),b(Ω,E),β) the space C∞
+P (∂),b(Ω,E)
+equipped with the topology β induced by the seminorms (∣ ⋅ ∣ν,α)ν∈C0(Ω),α∈A. The
+topology β is called the strict topology. It is a bit tricky to prove the ε-compatibility
+
+54
+4. CONSISTENCY
+of (C∞
+P (∂),b(Ω),β) and (C∞
+P (∂),b(Ω,E),β) because (C∞
+P (∂),b(Ω),β) may not be bar-
+relled.
+4.2.23. Remark. Let Ω ⊂ Rd be open and P(∂)K a hypoelliptic linear partial
+differential operator. Then (C∞
+P (∂),b(Ω),β) is non-barrelled if τc does not coincide
+with the ∥⋅∥∞-topology by [46, Section I.1, 1.15 Proposition, p. 12], e.g. (C∞
+∂,b(D),β)
+is non-barrelled.
+Hence we cannot use Proposition 3.1.11 c) directly.
+4.2.24. Proposition. Let Ω ⊂ Rd be open, P(∂)K a hypoelliptic linear partial
+differential operator and E an lcHs. Then (C∞
+P (∂),b(Ω),β)εE ≅ (C∞
+P (∂),b(Ω,E),β) if
+E has metric ccp.
+Proof. We set AP(Ω,E) ∶= C∞
+P (∂),b(Ω,E) and observe that (idEΩ,idΩK) is the
+generator of ((C∞
+P (∂),b(Ω),β),E). First, we prove that the generator is consistent.
+Clearly, we only need to show that S(u) ∈ AP(Ω,E) for every u ∈ (C∞
+P (∂),b(Ω),β)εE.
+Let u ∈ (C∞
+P (∂),b(Ω),β)εE. Next, we show that u ∈ CW∞
+P (∂)(Ω)εE with CW∞
+P (∂)(Ω)
+from Example 3.1.9 b). For α ∈ A there are an absolutely convex, compact K ⊂
+(C∞
+P (∂),b(Ω),β) and C > 0 such that for all f ′ ∈ (C∞
+P (∂),b(Ω),β)′ it holds that
+pα(u(f ′)) ≤ C sup
+f∈K
+∣f ′(f)∣.
+(24)
+We denote by τc the topology of compact convergence on C∞
+P (∂),b(Ω), i.e. the topol-
+ogy of uniform convergence on compact subsets of Ω. From the compactness of K in
+(C∞
+P (∂),b(Ω),β) it follows that K is ∥ ⋅ ∥∞-bounded and τc-compact by [45, Proposi-
+tion 1 (viii), p. 586] since (C∞
+P (∂),b(Ω),β) carries the induced topology of (Cb(Ω),β)
+and the strict topology β is the mixed topology γ(τc,∥ ⋅ ∥∞) by [45, Proposition 3,
+p. 590]. Let f ′ ∈ (C∞
+P (∂)(Ω),τc)′. Then there are M ⊂ Ω compact and C0 > 0 such
+that
+∣f ′(f)∣ ≤ C0 sup
+x∈M
+∣f(x)∣
+for all f ∈ C∞
+P (∂)(Ω). Choosing a compactly supported cut-off function ν ∈ C∞
+c (Ω)
+with ν = 1 near M, we obtain
+∣f ′(f)∣ ≤ C0 sup
+x∈Ω
+∣f(x)∣∣ν(x)∣ = C0∣f∣ν
+for all f ∈ C∞
+P (∂)(Ω).
+Therefore f ′ ∈ (C∞
+P (∂)(Ω),β)′.
+In combination with the
+τc-compactness of K it follows from (24) that u ∈ (C∞
+P (∂)(Ω),τc)εE. Using that
+(C∞
+P (∂)(Ω),τc) = CW∞
+P (∂)(Ω) as locally convex spaces by the hypoellipticity of
+P(∂)K (see e.g. [70, p. 690]), we obtain that u ∈ CW∞
+P (∂)(Ω)εE. Due to Propo-
+sition 3.1.11 c) this yields that S(u) ∈ C∞
+P (∂)(Ω,E). Furthermore, we note that
+∥S(u)∥∞,α = sup
+x∈Ω
+pα(S(u)(x)) = sup
+x∈Ω
+pα(u(δx)) ≤
+(24) C sup
+x∈Ω
+sup
+f∈K
+∣δx(f)∣
+= C sup
+f∈K
+∥f∥∞ < ∞
+as K is ∥ ⋅ ∥∞-bounded, implying that S(u) ∈ C∞
+P (∂),b(Ω,E) = AP(Ω,E). Hence the
+generator (idEΩ,idΩK) is consistent.
+It is easily seen that e′ ○ f ∈ C∞
+P (∂),b(Ω) = AP(Ω) for all e′ ∈ E′ and f ∈
+C∞
+P (∂),b(Ω,E) (see the proof of Proposition 3.1.10), which proves that the gener-
+ator is strong as well.
+Moreover, we define Nν(f) ∶= {f(x)∣ν(x)∣ ∣ x ∈ Ω} for
+f ∈ (C∞
+P (∂),b(Ω,E),β) and ν ∈ C0(Ω). The set K ∶= acx(Nν(f)) is absolutely con-
+vex and compact in E by Proposition A.1.3 (ii) because f∣ν∣ ∈ C0(Ω,E) and Ω
+second-countable, yielding our statement by Corollary 3.2.5 (iv).
+□
+
+4.2. FURTHER EXAMPLES OF ε-PRODUCTS
+55
+If Ω ⊂ C is an open, simply connected set, P(∂) = ∂ and E is complete, then
+the preceding result is also a consequence of [17, 3.10 Satz, p. 146].
+Next, we consider the vector-valued Beurling–Björck space Sµ(Rd,E) which
+generalises the Schwartz space and whose scalar-valued counterpart was studied by
+Björck in [20], by Schmeisser and Triebel in [155] (see [20, Definition 1.8.1, p. 375],
+[155, 1.2.1.2 Definition, p. 15]) whereas semigroups on its toplogical dual space
+were treated by Alvarez et al. in [5]. Since Fourier transformation is involved in the
+definition of Sµ(Rd,E), we start with the following statement.
+4.2.25. Proposition. Let E be a locally complete lcHs over C, f ∈ S(Rd,E)
+and x ∈ Rd. Then fe−i⟨x,⋅⟩ is Pettis-integrable on Rd where ⟨⋅,⋅⟩ is the usual scalar
+product on Rd.
+Proof. We choose m ∶= d + 1 and set ψ∶Rd → [0,∞), ψ(ζ) ∶= (1 + ∣ζ∣2)−m/2, as
+well as g∶Rd → [0,∞), g(ζ) ∶= ψ(ζ)−1. Then ψ ∈ L1(Rd,λ) and ψg = 1. Moreover,
+let x = (xi) ∈ Rd and set u∶Rd → E, u(ζ) ∶= f(ζ)e−i⟨x,ζ⟩g(ζ). We note that
+(∂en)Eu(ζ)
+= (∂en)Ef(ζ)e−i⟨x,ζ⟩g(ζ) − ixnf(ζ)e−i⟨x,ζ⟩g(ζ) + mf(ζ)e−i⟨x,ζ⟩(1 + ∣ζ∣2)(m/2)−1ζn
+for all ζ = (ζi) ∈ Rd and 1 ≤ n ≤ d, which implies
+pα((∂en)Eu(ζ)) ≤ pα((∂en)Ef(ζ))g(ζ) + ∣xn∣pα(f(ζ))g(ζ) + mpα(f(ζ))g(ζ)
+for all α ∈ A and hence
+sup
+ζ∈Rd
+β∈Nd
+0,∣β∣≤1
+pα((∂β)Eu(ζ)) ≤ (1 + ∣xn∣ + m)∣f∣S(Rd),m,α.
+Therefore u = fe−i⟨x,⋅⟩g is (weakly) C1
+b , which yields u ∈ C[1]
+b (Rd,E) by Proposition
+A.1.5. Now, we choose h∶Rd → (0,∞), h(ζ) ∶= 1 + ∣ζ∣2. Then
+sup
+ζ∈Rd pα(u(ζ)h(ζ)) ≤ sup
+ζ∈Rd pα(f(ζ))(1 + ∣ζ∣2)(m+2)/2 ≤ ∣f∣S(Rd),m+2,α < ∞
+for all α ∈ A, and for every ε > 0 there is r > 0 such that 1 ≤ εh(ζ) for all ζ ∉ Br(0) =∶
+K.
+We deduce from Proposition A.2.7 (iii) that fe−i⟨x,⋅⟩ is Pettis-integrable on
+Rd.
+□
+Thus, for f ∈ S(Rd,E) with locally complete E the Fourier transformation
+FE(f)∶Rd → E, FE(f)(x) ∶= (2π)−d/2 ∫
+Rd
+f(ζ)e−i⟨x,ζ⟩dζ,
+is defined. From the Pettis-integrability we get (e′ ○ FE)(f) = FC(e′ ○ f) for every
+e′ ∈ E′. As FC(e′ ○ f) ∈ S(Rd) for every e′ ∈ E′ by [20, Proposition 1.8.2, p. 375],
+we obtain from the weak-strong principle Corollary 5.2.21 (or [30, Theorem 9, p.
+232] and [131, Mackey’s theorem 23.15, p. 268]) that FE(f) ∈ S(Rd,E).
+For a locally complete lcHs E over C and a continuous function µ∶Rd → [0,∞)
+such that
+(γ) there are a ∈ R, b > 0 with µ(x) ≥ a + bln(1 + ∣x∣) for all x ∈ Rd,
+we set
+Sµ(Rd,E) ∶= {f ∈ C∞(Rd,E) ∣ ∀ m,j ∈ N0, α ∈ A ∶ ∣f∣j,m,α < ∞}
+where ∣f∣m,j,α ∶= max(qm,j,α(f),qm,j,α(FE(f))) with
+qm,j,α(f) ∶=
+sup
+x∈Rd
+β∈Nd
+0,∣β∣≤m
+pα((∂β)Ef(x))ejµ(x).
+
+56
+4. CONSISTENCY
+We note that from qm,j,α(f) < ∞ for all m,j ∈ N0, α ∈ A and condition (γ) it follows
+that f ∈ S(Rd,E) and hence qm,j,α(FE(f)) is defined. Further, we set Sµ(Rd) ∶=
+Sµ(Rd,C). We observe that Sµ(Rd,E) is a dom-space. Indeed, let ωm ∶= ̃ωm ∪ ̃ωm,1
+where ̃ωm ∶= Mm × Rd with Mm ∶= {β ∈ Nd
+0 ∣ ∣β∣ ≤ m} and ̃ωm,1 ∶= ̃ωm × {1} for all
+m ∈ N0. Setting domT E
+m ∶= S(Rd,E) and T E
+m∶S(Rd,E) → Eωm by
+T E
+m(f)(β,x) ∶= (∂β)Ef(x) and T E
+m(f)(β,x,1) ∶= ((∂β)E ○ FE)f(x),
+(β,x) ∈ ̃ωm,
+for every m ∈ N0 as well as AP(Rd,E) ∶= ERd, we have that Sµ(Rd,E) is a dom-
+space with weights given by νj,m(β,x) ∶= νj,m(β,x,1) ∶= ejµ(x) for all (β,x) ∈ ̃ωm
+and m,j ∈ N0.
+The condition (γ) is introduced in [20, p. 363]. Choosing µ(x) ∶= ln(1 + ∣x∣),
+x ∈ Rd, we get the Schwartz space Sµ(Rd,E) = S(Rd,E) back.
+4.2.26. Example. Let E be a locally complete lcHs over C and µ∶Rd → [0,∞)
+continuous such that condition (γ) is fulfilled.
+(i) If E has metric ccp, or
+(ii) if µ ∈ C1(Rd) and there are k ∈ N0, C > 0 such that ∣∂enµ(x)∣ ≤ Cekµ(x)
+for all x ∈ Rd and 1 ≤ n ≤ d,
+then Sµ(Rd,E) ≅ Sµ(Rd)εE.
+Proof. First, we show that the generator (T E
+m,T C
+m)m∈N0 for (Sµ,E) is strong
+and consistent. From
+(∂β)C(e′ ○ f)(x) = e′((∂β)Ef(x)),
+(β,x) ∈ ̃ωm,
+where ̃ωm = {β ∈ Nd
+0 ∣ ∣β∣ ≤ m} × Rd, we get in combination with the Pettis-
+integrability by Proposition 4.2.25 that
+((∂β)C ○ FC)(e′ ○ f)(x) = e′(((∂β)E ○ FE)f(x)),
+(β,x) ∈ ̃ωm
+(25)
+for all e′ ∈ E′, f ∈ Sµ(Rd,E) and m ∈ N0, which means that the generator is
+strong. For consistency we consider the case µ(x) = ln(1 + ∣x∣), x ∈ Rd, i.e. the
+Schwartz space, first. Due to Corollary 3.2.10 the map S∶S(Rd)εE → S(Rd,E) is
+an isomorphism and according to Theorem 3.2.4 its inverse is given by
+Rt∶S(Rd,E) → S(Rd)εE, f ↦ J −1 ○ Rt
+f.
+Let u ∈ S(Rd)εE. Thanks to the proof of Corollary 3.2.10 we only need to show
+that
+u(δx ○ (∂β ○ FC)) = (∂β)EFE(S(u))(x),
+x ∈ Rd.
+We set f ∶= S(u) ∈ S(Rd,E) and from (25) we obtain
+Rt
+f(δx ○ ((∂β)C ○ FC))(e′) = (∂β)CFC(e′ ○ f)(x) = e′((∂β)EFE(f)(x)),
+e′ ∈ E′,
+for all x ∈ Rd and β ∈ Nd
+0, which results in
+u(δx ○ ((∂β)C ○ FC)) = S−1(f)(δx ○ ((∂β)C ○ FC)) = J −1(Rt
+f(δx ○ ((∂β)C ○ FC)))
+= (∂β)EFE(f)(x) = (∂β)EFE(S(u))(x).
+(26)
+Thus (T E
+m,T C
+m)m∈N0 is a consistent generator for (S,E).
+Let us turn to general µ. Let u ∈ Sµ(Rd)εE. We show that u ∈ S(Rd)εE. Then
+it follows from the first part of the proof that (T E
+m,T C
+m)m∈N0 is a consistent generator
+for (Sµ,E). For α ∈ A there are an absolutely convex compact set K ⊂ Sµ(Rd) and
+C > 0 such that for all f ′ ∈ Sµ(Rd)′ it holds
+pα(u(f ′)) ≤ C sup
+f∈K
+∣f ′(f)∣.
+(27)
+
+4.2. FURTHER EXAMPLES OF ε-PRODUCTS
+57
+The compactness of K in Sµ(Rd) and the estimate
+sup
+x∈Rd
+β∈Nd
+0,∣β∣≤m
+∣(∂β)Cf(x)∣(1 + ∣x∣2)j/2 ≤
+sup
+x∈Rd
+β∈Nd
+0,∣β∣≤m
+∣(∂β)Cf(x)∣e(j/2)(2/b)(µ(x)−a)
+≤ e−(aj)/b∣f∣j,m,
+f ∈ Sµ(Rd),
+for all j,m ∈ N0 by condition (γ) imply that the inclusion Sµ(Rd) ↪ S(Rd) is
+continuous and thus that K is compact in S(Rd). Let f ′ ∈ S(Rd)′. Then there are
+j,m ∈ N0 and C0 > 0 such that
+∣f ′(f)∣ ≤ C0
+sup
+x∈Rd
+β∈Nd
+0,∣β∣≤m
+∣(∂β)Cf(x)∣(1 + ∣x∣2)j/2 ≤ C0e−(aj)/b∣f∣j,m
+for all f ∈ Sµ(Rd). Hence f ′ ∈ Sµ(Rd)′ and from (27) we obtain that u ∈ S(Rd)εE
+because K is absolutely convex and compact in S(Rd).
+Condition (γ) implies that µ(x) → ∞ for ∣x∣ → ∞. Noting that for every j ∈ N
+and ε > 0 there is r > 0 such that
+ejµ(x)
+e2jµ(x) = e−jµ(x) < ε
+(28)
+for all x ∉ Br(0), we deduce ∣f∣Rd∖Br(0),m,j,α ≤ ε∣f∣m,2j,α for every f ∈ Sµ(Rd,E),
+m ∈ N0 and α ∈ A.
+(i) Thus, if E has metric ccp, then the sets Kβ ∶= acx((∂β)Efejµ(Rd)) and
+Kβ,1 ∶= acx((∂β)EFE(f)ejµ(Rd)) are absolutely convex and compact by Propo-
+sition A.1.3 (ii) for every f ∈ Sµ(Rd,E), j,m ∈ N0 and β ∈ Mm as (∂β)Efejµ ∈
+C0(Rd,E) and (∂β)EFE(f)ejµ ∈ C0(Rd,E).
+(ii) We set g0∶Rd → E, g0(x) ∶= (∂β)Ef(x)ejµ(x), and g1∶Rd → E, g1(x) ∶=
+(∂β)EFE(f)(x)ejµ(x), for j,m ∈ N0 and β ∈ Mm. We observe that
+(∂en)Eg0(x) = (∂β+en)Ef(x)ejµ(x) + j(∂β)Ef(x)ejµ(x)∂enµ(x)
+and
+(∂en)Eg1(x) = (∂β+en)EFE(f)(x)ejµ(x) + j(∂β)EFE(f)(x)ejµ(x)∂enµ(x)
+for all x ∈ Rd and 1 ≤ n ≤ d. As in Example 3.2.9 it follows from condition (ii) that
+there are k ∈ N0, C > 0 such that
+sup
+x∈Rd
+γ∈Nd
+0,∣γ∣≤1
+pα((∂γ)Egi(x)) ≤ ∣f∣m+1,j,α + Cj∣f∣m,j+k,α
+for all α ∈ A and i = 0,1. Thus g0 and g1 are (weakly) C1
+b . We set h ∶= ejµ and note
+that
+sup
+x∈Rd pα(gi(x)h(x)) ≤ ∣f∣m,2j,α < ∞
+for all α ∈ A and i = 0,1. This yields that Kβ = acx(g0(Rd)) and Kβ,1 = acx(g1(Rd))
+are absolutely convex and compact by Proposition A.1.4 with (28) and Proposition
+A.1.5.
+Then we have in both cases
+Nj,m(f) = ({(∂β)Ef(x)ejµ(x) ∣ x ∈ Rd, β ∈ Mm}
+∪ {(∂β)EFE(f)(x)ejµ(x) ∣ x ∈ Rd, β ∈ Mm})
+⊂acx( ⋃
+β∈Mm
+(Kβ ∪ Kβ,1))
+
+58
+4. CONSISTENCY
+and the set on the right-hand side is absolutely convex and compact by [89, 6.7.3
+Proposition, p. 113], which implies that Sµ(Rd,E) ≅ Sµ(Rd)εE by Corollary 3.2.5
+(iv).
+□
+We come back to these spaces in Theorem 5.1.5.
+Another example that is
+related to Fourier transformation is the space of vector-valued smooth functions
+that are 2π-periodic in each variable. We equip the space C∞(Rd,E) for an lcHs E
+with the system of seminorms generated by
+∣f∣K,m,α ∶=
+sup
+x∈Rd
+β∈Nd
+0,∣β∣≤m
+pα((∂β)Ef(x))χK(x) =
+sup
+x∈K
+β∈Nd
+0,∣β∣≤m
+pα((∂β)Ef(x)),
+f ∈ C∞(Rd,E),
+for K ⊂ Rd compact, m ∈ N0 and α ∈ A, i.e. we consider CW∞(Rd,E).
+By
+C∞
+2π(Rd,E) we denote the topological subspace of C∞(Rd,E) consisting of the func-
+tions which are 2π-periodic in each variable. Further, we set C∞
+2π(Rd) ∶= C∞
+2π(Rd,K).
+4.2.27. Example. If E is a locally complete lcHs, then C∞
+2π(Rd,E) ≅ C∞
+2π(Rd)εE.
+Proof. First, we note that for each x ∈ Rd and 1 ≤ n ≤ d we have δx = δx+2πen
+in C∞
+2π(Rd)′ and thus
+SC∞
+2π(Rd)(u)(x) − SC∞
+2π(Rd)(u)(x + 2πen) = u(δx − δx+2πen) = 0,
+u ∈ C∞
+2π(Rd)εE,
+implying that SC∞
+2π(Rd)(u) is 2π-periodic in each variable. In addition, we observe
+that e′ ○ f is 2π-periodic in each variable for all e′ ∈ E′ and f ∈ C∞
+2π(Rd,E). Now,
+we obtain as in Example 3.2.8 a) for k = ∞ that SC∞
+2π(Rd)∶C∞
+2π(Rd)εE → C∞
+2π(Rd,E)
+is an isomorphism.
+□
+We return to C∞
+2π(Rd,E) in Theorem 5.1.4 and Theorem 5.6.14. Now, we direct
+our attention to spaces of continuously partially differentiable functions on an open
+bounded set such that all derivatives can be continuously extended to the boundary.
+Let E be an lcHs, k ∈ N∞ and Ω ⊂ Rd open and bounded. The space Ck(Ω,E) is
+given by
+Ck(Ω,E) ∶= {f ∈ Ck(Ω,E) ∣ (∂β)Ef cont. extendable on Ω for all β ∈ Nd
+0, ∣β∣ ≤ k}
+and equipped with the system of seminorms given by
+∣f∣α ∶=
+sup
+x∈Ω
+β∈Nd
+0,∣β∣≤k
+pα((∂β)Ef(x)),
+f ∈ Ck(Ω,E),
+for α ∈ A if k < ∞, and by
+∣f∣m,α ∶=
+sup
+x∈Ω
+β∈Nd
+0,∣β∣≤m
+pα((∂β)Ef(x)),
+f ∈ C∞(Ω,E),
+for m ∈ N0 and α ∈ A if k = ∞. Further, we set Ck(Ω) ∶= Ck(Ω,K).
+4.2.28. Example. Let E be an lcHs, k ∈ N∞ and Ω ⊂ Rd open and bounded.
+(i) If E has metric ccp, or
+(ii) if E is locally complete, k = ∞ and there exists C > 0 such that for each
+x,y ∈ Ω there is a continuous path from x to y in Ω whose length is
+bounded by C∣x − y∣,
+then Ck(Ω,E) ≅ Ck(Ω)εE.
+Proof. The generator coincides with the one of Example 4.2.22.
+Due to
+Proposition 3.1.11 we have S(u) ∈ Ck(Ω,E) and
+(∂β)ES(u)(x) = u(δx ○ (∂β)K),
+β ∈ Nd
+0, ∣β∣ ≤ k, x ∈ Ω,
+
+4.3. RIESZ–MARKOV–KAKUTANI REPRESENTATION THEOREMS
+59
+for all u ∈ Ck(Ω)εE since Ck(Ω) is a Banach space if k < ∞, and a Fréchet space
+if k = ∞, in particular, both are barrelled. As a consequence of Proposition 4.1.7
+and Lemma 4.1.8 with T = (∂β)K for β ∈ Nd
+0, ∣β∣ ≤ k, we obtain that (∂β)ES(u) ∈
+Cext(Ω,E) for all u ∈ Ck(Ω)εE. Thus the generator is consistent. It is easy to check
+that it is strong, too. This yields (ii) by Corollary 3.2.5 (ii) since C∞(Ω) is a nuclear
+Fréchet space by [131, Examples 28.9 (5), p. 350] under the conditions on Ω.
+Let us turn to part (i). Let f ∈ Ck(Ω,E), J ∶= {1}, m ∈ N0 and set Mm ∶= {β ∈
+Nd
+0 ∣ ∣β∣ ≤ k} if k < ∞, and Mm ∶= {β ∈ Nd
+0 ∣ ∣β∣ ≤ m} if k = ∞. We denote by fβ the
+continuous extension of (∂β)Ef on the compact metrisable set Ω. The set
+N1,m(f) = {(∂β)Ef(x) ∣ x ∈ Ω, β ∈ Mm} ⊂
+⋃
+β∈Mm
+fβ(Ω)
+is relatively compact and metrisable since it is a subset of a finite union of the
+compact metrisable sets fβ(Ω) as in Example 3.2.8. Due to Corollary 3.2.5 (iv) we
+obtain our statement (i) as E has metric ccp.
+□
+We close this section by an examination of the topological subspace
+E0(E) ∶= {f ∈ C∞([0,1],E) ∣ ∀ k ∈ N0 ∶ (∂k)Ef(1) = 0}
+where (∂k)Ef(1) ∶= limx→1+(∂k)Ef(x). Further, we set E0 ∶= E0(K).
+4.2.29. Example. Let E be a locally complete lcHs. Then E0εE ≅ E0(E).
+Proof. We note that Ω ∶= (0,1) satisfies the condition on Ω in Example 4.2.28
+(ii) with C ∶= 1 and thus C∞([0,1]) and its closed subspace E0 are nuclear Fréchet
+spaces. The generator coincides with the one of Example 4.2.28. From the proof of
+Example 4.2.28 we know that
+lim
+x→1+(∂k)ES(u)(x) = u(δ1 ○ (∂k)K) = u(0) = 0,
+k ∈ N0,
+for all u ∈ E0εE. In combination with Example 4.2.28 this yields the consistency
+of the generator. Again, its strength is easy to check. Therefore our statement is
+valid by Corollary 3.2.5 (ii).
+□
+4.3. Riesz–Markov–Kakutani representation theorems
+In this subsection we generalise the concept of strength and consistency such
+that it is not strictly bounded to dom-spaces and their generators anymore. This
+allows us to answer the question: Given T K
+m ∈ F(Ω)′ is there T E
+m ∈ L(F(Ω,E),E)
+such that (T E
+m,T K
+m) is strong and consistent? Furthermore, we will see that the
+operators T E
+m are usually the ones that can be obtained from integral representations
+of T K
+m, i.e. we transfer Riesz–Markov–Kakutani theorems from the scalar-valued to
+the vector-valued case.
+We recall that the Riesz–Markov–Kakutani theorem for
+compact topological Hausdorff spaces Ω says that for every T R ∈ Cb(Ω)′ there is a
+unique regular R-valued Borel measure µ on Ω such that
+T R(f) = ∫
+Ω
+f(x)dµ(x),
+f ∈ Cb(Ω),
+(29)
+which was proved by Riesz [146, p. 976] in the case Ω ∶= [0,1] and by Kakutani [95,
+Theorem 9, p. 1009] for general compact Hausdorff Ω (see Saks [152, Eq. (1.1), 6., p.
+408, 411] for compact metric Ω). Markov treated the case where Ω is a normal (not
+necessarily Hausdorff) topological space and the T R are positive linear functionals
+on Cb(Ω) such that T R(1) = 1 [127, Definition 2, p. 167]. In this case, for every
+such T R there is a unique exterior density µ on Ω in the sense of [127, Definition
+3, p. 167] such that (29) holds by [127, Theorem 22, p. 184] and the right-hand
+side is read in the sense of [127, Eq. (71), (72), (80), p. 180–181] (see also [60,
+
+60
+4. CONSISTENCY
+IV.6.2 Theorem, p. 262] for a more familiar version with regular (finitely) additive
+bounded Borel measures µ).
+4.3.1. Definition (strong, consistent). Let E be an lcHs and Ω a non-empty
+set.
+Let F(Ω) ⊂ KΩ and F(Ω,E) ⊂ EΩ be lcHs such that δx ∈ F(Ω)′ for all
+x ∈ Ω.
+Let (ωm)m∈M be a family of non-empty sets, T K
+m∶domT K
+m → Kωm and
+T E
+m∶domT E
+m → Eωm be linear with F(Ω) ⊂ domT K
+m ⊂ KΩ and F(Ω,E) ⊂ domT E
+m ⊂
+EΩ for all m ∈ M.
+a) We call (T E
+m,T K
+m)m∈M a consistent family for (F(Ω),E), in short (F,E),
+if we have for every u ∈ F(Ω)εE, m ∈ M and x ∈ ωm that
+(i) S(u) ∈ F(Ω,E) and T K
+m,x ∶= δx ○ T K
+m ∈ F(Ω)′,
+(ii) T E
+mS(u)(x) = u(T K
+m,x).
+b) We call (T E
+m,T K
+m)m∈M a strong family for (F(Ω),E), in short (F,E), if
+we have for every e′ ∈ E′, f ∈ F(Ω,E), m ∈ M and x ∈ ωm that
+(i) e′ ○ f ∈ F(Ω),
+(ii) T K
+m(e′ ○ f)(x) = (e′ ○ T E
+m(f))(x).
+Note that ωm need not be a subset of Ω. As a convention we omit the index m
+of the set ωm, the operators T E
+m and T K
+m if M is a singleton. The following remark
+shows that the preceding definition of a consistent resp. strong family coincides with
+the usual definition in the case of generators of dom-spaces (see Definition 3.1.7).
+4.3.2. Remark. Let (T E
+m,T K
+m)m∈M be a generator for (FV,E). We note that
+the condition T K
+m,x ∈ FV(Ω)′ for all m ∈ M and x ∈ ωm in a)(i) of Definition
+4.3.1 is always satisfied for generators by Remark 3.1.6 b). Moreover, if S(u) ∈
+AP(Ω,E) ∩ domT E
+m for u ∈ FV(Ω)εE and all m ∈ M and a)(ii) of Definition 4.3.1
+is fulfilled, then S(u) ∈ FV(Ω,E) by Lemma 3.1.8, implying that a)(i) is satisfied.
+Further, if f ∈ FV(Ω,E) and e′ ○f ∈ AP(Ω)∩domT K
+m for all e′ ∈ E′ and m ∈ M and
+b)(ii) of Definition 4.3.1 is fulfilled, then e′ ○ f ∈ FV(Ω) by Lemma 3.2.1, implying
+that b)(i) is satisfied.
+The next proposition is the key result in transferring Riesz–Markov–Kakutani
+theorems from the scalar-valued to the vector-valued case. To state this proposition
+we need that our map S∶F(Ω)εE → F(Ω,E) is an isomorphism and that its inverse
+is given as in Theorem 3.2.4, i.e. that
+Rt∶F(Ω,E) → F(Ω)εE, f ↦ J −1 ○ Rt
+f,
+is the inverse of S where Rt
+f(f ′)(e′) = f ′(e′ ○ f), for f ′ ∈ F(Ω)′ and e′ ∈ E′, and
+J ∶E → E′⋆ is the canonical injection in the algebraic dual E′⋆ of E′.
+4.3.3. Proposition. Let E be an lcHs, (Ω,Σ,µ) a measure space and F(Ω)
+and F(Ω,E) ε-compatible with inverse Rt of S and (T E
+0 ,T K
+0 ) a strong family for
+(F,E) with ω0 ∶= Ω.
+If T K
+0 (f) is integrable for every f ∈ F(Ω) and T E
+0 (f) is
+Pettis-integrable on Ω for every f ∈ F(Ω,E) and
+T K∶F(Ω) → K, T K(f) ∶= ∫
+Ω
+T K
+0 (f)(x)dµ(x),
+is continuous, then
+u(T K) = ∫
+Ω
+T E
+0 S(u)(x)dµ(x),
+u ∈ F(Ω)εE.
+Proof. Let u ∈ F(Ω)εE and set f ∶= S(u) ∈ F(Ω,E). We have
+Rt
+f(T K)(e′) = T K(e′○f) = ∫
+Ω
+T K
+0 (e′○f)(x)dµ(x) = ⟨e′,∫
+Ω
+T E
+0 f(x)dµ(x)⟩,
+e′ ∈ E′,
+
+4.3. RIESZ–MARKOV–KAKUTANI REPRESENTATION THEOREMS
+61
+by the strength of (T E
+0 ,T K
+0 ) and the Pettis-integrability of T E
+0 (f), which yields
+u(T K) = S−1(f)(T K) = J −1(Rt
+f(T K)) = ∫
+Ω
+T E
+0 f(x)dµ(x) = ∫
+Ω
+T E
+0 S(u)(x)dµ(x)
+due to Rt being the inverse of S.
+□
+4.3.4. Proposition. Let E be an lcHs, (Ω,Σ,µ) a measure space, (T E
+0 ,T K
+0 ) a
+strong family for (F,E) with ω0 ∶= Ω such that T E
+0 (f) is Pettis-integrable on Ω for
+every f ∈ F(Ω,E), and (T E,T K) a consistent family for (F,E) such that
+T E(f) = ∫
+Ω
+T E
+0 f(x)dµ(x),
+f ∈ F(Ω,E).
+Then (T E,T K) is a strong family for (F,E), T K
+0 (f) is integrable for every f ∈ F(Ω)
+and
+T K = ∫
+Ω
+T K
+0 (f)(x)dµ(x),
+f ∈ F(Ω).
+Proof. We set f ⋅ e∶Ω → E, (f ⋅ e)(x) ∶= f(x)e, for e ∈ E and f ∈ F(Ω). Since
+(T E,T K) is a consistent family for (F,E), we get f ⋅e = S(Θ(e⊗f)) ∈ F(Ω,E) and
+T E(f ⋅ e) = Θ(e ⊗ f)(T K) = T K(f)e
+(30)
+with the map Θ from (3). From the strength of (T E
+0 ,T K
+0 ) we deduce that
+e′ ○ T E
+0 (f ⋅ e) = T K
+0 (e′ ○ (f ⋅ e)) = T K
+0 (e′(e)f) = e′(e)T K
+0 (f)
+and from the Pettis-integrability of T E
+0 (f ⋅ e) that
+T K(f)e′(e) =
+(30) ⟨e′,T E(f ⋅ e)⟩ = ∫
+Ω
+e′(e)T K
+0 (f)(x)dµ(x)
+for all e′ ∈ E′. This implies that e′(e)T K
+0 (f) is integrable for all e′ ∈ E′. Further,
+since E is non-trivial by our assumptions in Chapter 2, there is some e0 ∈ E, e0 ≠ 0.
+By the Hahn–Banach theorem there is some e′
+0 ∈ E′ with e′
+0(e0) ≠ 0, which yields
+that T K
+0 (f) is integrable and
+T K(f) = ∫
+Ω
+T K
+0 (f)(x)dµ(x).
+Furthermore, we conclude in combination with the strength of (T E
+0 ,T K
+0 ) and the
+Pettis-integrability of T E
+0 (f) for all f ∈ F(Ω,E) that
+⟨e′,T E(f)⟩ = ∫
+Ω
+T K
+0 (e′ ○ f)(x)dµ(x) = T K(e′ ○ f)
+for all f ∈ F(Ω,E) and e′ ∈ E′, which means that (T E,T K) is a strong family for
+(F,E).
+□
+Let us apply the preceding propositions to the space D([0,1],E) of E-valued
+càdlàg functions on [0,1]. For f ∈ D([0,1],E) we set f(x−) ∶= limw→x− f(w) if
+x ∈ (0,1], and f(0−) ∶= 0.
+4.3.5. Proposition. Let E be a quasi-complete lcHs. Then for every T K ∈
+D([0,1])′ there is T E ∈ L(D([0,1],E),E) such that (T E,T K) is a consistent family
+for (D,E) and there are a unique regular K-valued Borel measure µ on [0,1] and
+a unique ϕ ∈ ℓ1([0,1],K) such that
+T E(f) = ∫
+[0,1]
+f(x)dµ(x) +
+∑
+x∈[0,1]
+(f(x) − f(x−))ϕ(x),
+f ∈ D([0,1],E).
+(31)
+
+62
+4. CONSISTENCY
+On the other hand, if (T E,T K) is a consistent family, there is a unique regular K-
+valued Borel measure µ on [0,1] such that (31) holds and T E ∈ L(D([0,1],E),E).
+Proof. Due to the representation theorem [139, Theorem 1, p. 383] there are
+a unique regular K-valued Borel measure µ on [0,1] and a unique ϕ ∈ ℓ1([0,1],K)
+such that
+T K(f) = ∫
+[0,1]
+f(x)dµ(x) +
+∑
+x∈[0,1]
+(f(x) − f(x−))ϕ(x),
+f ∈ D([0,1],K).
+(32)
+By Example 4.2.5 S∶D([0,1])εE → D([0,1],E) is an isomorphism with inverse
+Rt∶f ↦ J ○ Rt
+f.
+The next part is the analogon of Proposition 4.3.3 for ∑x∈[0,1]. We set
+T K
+1 ∶D([0,1]) → K, T K
+1 (f) ∶=
+∑
+x∈[0,1]
+(f(x) − f(x−))ϕ(x),
+and note that T K
+1 ∈ D([0,1])′. Let u ∈ D([0,1])εE and set f ∶= S(u) ∈ D([0,1],E).
+We have
+Rt
+f(T K
+1 )(e′) = T K
+1 (e′ ○ f) =
+∑
+x∈[0,1]
+((e′ ○ f)(x) − (e′ ○ f)(x−))ϕ(x)
+= ⟨e′,
+∑
+x∈[0,1]
+(f(x) − f(x−))ϕ(x)⟩,
+e′ ∈ E′,
+due to the Pettis-summability of x ↦ (f(x) − f(x−))ϕ(x) on [0,1] by Proposition
+A.2.6, which yields
+u(T K
+1 ) = S−1(f)(T K
+1 ) = J −1(Rt
+f(T K
+1 )) =
+∑
+x∈[0,1]
+(f(x) − f(x−))ϕ(x)
+=
+∑
+x∈[0,1]
+(S(u)(x) − S(u)(x−))ϕ(x).
+(33)
+We note that every f ∈ D([0,1],E) is Pettis-integrable and that x ↦ (f(x) ��
+f(x−))ϕ(x) is Pettis-summable on [0,1] by Proposition A.2.6. Further,
+pα( ∫
+[0,1]
+f(x)dµ(x)+ ∑
+x∈[0,1]
+(f(x)−f(x−))ϕ(x)) ≤ (∣µ∣([0,1])+2∥ϕ∥ℓ1) sup
+x∈[0,1]
+pα(f(x))
+for all f ∈ D([0,1],E) and α ∈ A. The rest follows from Proposition 4.3.3 with
+(T E
+0 ,T K
+0 ) ∶= (idE[0,1],idK[0,1]) combined with (33). For the uniqueness of µ in (31)
+use that the µ in (32) is unique and Proposition 4.3.4 (and for the uniqueness of ϕ
+use an analogon of Proposition 4.3.4 for (T E
+1 ,T K
+1 )).
+□
+Let us turn to continuous functions that vanish at infinity.
+4.3.6. Proposition. Let Ω be a locally compact [second countable] topological
+Hausdorff space and E an lcHs with [metric] ccp. Then for every T K ∈ C0(Ω)′ there
+is T E ∈ L(C0(Ω,E),E) such that (T E,T K) is a consistent family for (C0,E) and
+there is a unique regular K-valued Borel measure µ on Ω such that
+T E(f) = ∫
+Ω
+f(x)dµ(x),
+f ∈ C0(Ω,E).
+(34)
+On the other hand, if (T E,T K) is a consistent family, then there is a unique regular
+K-valued Borel measure µ on Ω such that (34) holds and T E ∈ L(C0(Ω,E),E).
+
+4.3. RIESZ–MARKOV–KAKUTANI REPRESENTATION THEOREMS
+63
+Proof. Due to the Riesz–Markov–Kakutani representation theorem (see [149,
+6.19 Theorem, p. 130]) there is a unique regular K-valued Borel measure µ on Ω
+such that
+T K(f) = ∫
+Ω
+f(x)dµ(x),
+f ∈ C0(Ω).
+(35)
+By Example 4.2.11 S∶C0(Ω)εE → C0(Ω,E) is an isomorphism with inverse Rt∶f ↦
+J ○ Rt
+f. We note that every f ∈ C0(Ω,E) is Pettis-integrable by Proposition A.2.7
+(i) resp. (ii) with ψ ∶= g ∶= 1 since
+∫
+Ω
+∣ψ(x)∣d∣µ∣(x) = ∣µ∣(Ω) < ∞
+and
+pα(∫
+Ω
+f(x)dµ(x)) ≤ ∣µ∣(Ω)sup
+x∈Ω
+pα(f(x)),
+f ∈ C0(Ω,E),
+for all α ∈ A. The rest follows from Proposition 4.3.3 with (T E
+0 ,T K
+0 ) ∶= (idEΩ,idKΩ).
+For the uniqueness of µ in (34) use that the µ in (35) is unique and Proposition
+4.3.4.
+□
+Next, we consider the space of bounded continuous E-valued functions on a
+locally compact topological Hausdorff space Ω, i.e.
+Cb(Ω,E) = {f ∈ C(Ω,E) ∣ ∀ α ∈ A ∶ sup
+x∈Ω
+pα(f(x)) < ∞},
+but equipped with the strict topology β (see Remark 4.2.23) which is induced by
+the seminorms
+∣f∣ν,α ∶= sup
+x∈Ω
+pα(f(x))∣ν(x)∣,
+f ∈ Cb(Ω,E),
+for ν ∈ C0(Ω) and α ∈ A.
+4.3.7. Proposition. Let Ω be a locally compact [second countable] topological
+Hausdorff space and E an lcHs with [metric] ccp. Then for every T K ∈ (Cb(Ω),β)′
+there is T E ∈ L((Cb(Ω,E),β),E) such that (T E,T K) is a consistent family for
+((Cb(Ω),β),E) and there is a unique regular K-valued Borel measure µ on Ω such
+that
+T E(f) = ∫
+Ω
+f(x)dµ(x),
+f ∈ Cb(Ω,E).
+(36)
+On the other hand, if (T E,T K) is a consistent family, then there is a unique regular
+K-valued Borel measure µ on Ω such that (36) holds and T E ∈ L((Cb(Ω,E),β),E).
+Proof. Due to the Riesz–Markov–Kakutani representation theorem [89, 7.6.3
+Theorem, p. 141] for the strict topology there is a unique regular K-valued Borel
+measure µ on Ω such that
+T K(f) = ∫
+Ω
+f(x)dµ(x),
+f ∈ Cb(Ω).
+Since T K is continuous, there are ν ∈ C0(Ω) and C > 0 such that
+∣∫
+Ω
+⟨e′,f(x)⟩dµ(x)∣ = ∣T K(e′ ○ f)∣ ≤ C sup
+x∈Ω
+∣(e′ ○ f)(x)ν(x)∣ ≤ C sup
+x∈K
+∣e′(x)∣,
+e′ ∈ E′,
+for f ∈ Cb(Ω,E) with K ∶= acx(fν(Ω)). As K is absolutely convex and compact by
+Proposition A.1.3, f is Pettis-integrable on Ω w.r.t. µ by the Mackey–Arens theo-
+rem. The remaining parts of the proof follow from Example 4.2.11 and Proposition
+4.3.3 as in Proposition 4.3.6.
+□
+
+64
+4. CONSISTENCY
+4.3.8. Proposition. Let Ω be a locally compact [second countable] topological
+Hausdorff space and E an lcHs with [metric] ccp. Then for every T K ∈ CW(Ω)′ there
+is T E ∈ L(CW(Ω,E),E) such that (T E,T K) is a consistent family for (CW,E) and
+there is a unique regular K-valued Borel measure µ on Ω with compact support such
+that
+T E(f) = ∫
+Ω
+f(x)dµ(x),
+f ∈ C(Ω,E).
+(37)
+On the other hand, if (T E,T K) is a consistent family, then there is a unique regular
+K-valued Borel measure µ on Ω with compact support such that (37) holds and
+T E ∈ L(CW(Ω,E),E).
+Proof. By the Riesz–Markov–Kakutani representation theorem given in the
+remark after [35, Chap. 4, §4.8, Proposition 14, p. INT IV.48] for the topology of
+compact convergence there is a unique regular K-valued Borel measure µ on Ω with
+compact support such that
+T K(f) = ∫
+Ω
+f(x)dµ(x),
+f ∈ C(Ω).
+Since T K is continuous, there are a compact set M ⊂ Ω and C > 0 such that
+∣∫
+Ω
+⟨e′,f(x)⟩dµ(x)∣ = ∣T K(e′ ○ f)∣ ≤ C sup
+x∈M
+∣(e′ ○ f)(x)∣ ≤ C sup
+x∈K
+∣e′(x)∣,
+e′ ∈ E′,
+for f ∈ C(Ω,E) with the absolutely convex and compact set K ∶= acx(f(M)),
+implying that f is Pettis-integrable on Ω w.r.t. µ by the Mackey–Arens theorem.
+The rest of the proof is identical to the one of Proposition 4.3.7.
+□
+4.3.9. Proposition. Let Ω ⊂ Rd be open and E a locally complete lcHs. Then
+for every T K ∈ CW∞(Ω)′ there is T E ∈ L(CW∞(Ω,E),E) such that (T E,T K) is
+a consistent family for (CW∞,E). Given any open neighbourhood U ⊂ Ω of the
+compact distributional support suppT K of T K there are m ∈ N0 and a family of
+K-valued Radon measures (µβ)β∈Nd
+0,∣β∣≤m on Ω such that suppµβ ⊂ U for all β ∈ Nd
+0,
+∣β∣ ≤ m, and
+T E(f) = ∑
+∣β∣≤m
+∫
+Ω
+(∂β)Ef(x)dµβ(x),
+f ∈ C∞(Ω,E).
+(38)
+On the other hand, if (T E,T K) is a consistent family, then given any open neigh-
+bourhood U ⊂ Ω of the compact distributional support suppT K of T K there are
+m ∈ N0 and a family of K-valued Radon measures (µβ)β∈Nd
+0,∣β∣≤m on Ω such that
+(38) holds, suppµβ ⊂ U for all β ∈ Nd
+0, ∣β∣ ≤ m and T E ∈ L(CW∞(Ω,E),E).
+Proof. T K ∈ CW∞(Ω)′ is a distribution with compact support and thus has
+finite order by [171, Corollary, p. 259]. Denote by m ∈ N0 the order of T K. Given
+any open neighbourhood U ⊂ Ω of suppT K there is a family of K-valued Radon
+measures (µβ)β∈Nd
+0,∣β∣≤m on Ω such that
+T K(f) = ∑
+∣β∣≤m
+∫
+Ω
+(∂β)Kf(x)dµβ(x),
+f ∈ C∞(Ω),
+and suppµβ ⊂ U for all β ∈ Nd
+0, ∣β∣ ≤ m by [171, Theorem 24.4, p. 259]. Since the
+support Kβ ∶= suppµβ of µβ is compact
+∫
+Ω
+(∂β)Kf(x)dµβ(x) = ∫
+Kβ
+(∂β)Kf(x)dµβ(x),
+f ∈ C∞(Ω),
+
+4.3. RIESZ–MARKOV–KAKUTANI REPRESENTATION THEOREMS
+65
+and (∂β)Ef ∈ C1(Ω,E) for f ∈ C∞(Ω,E), it follows from Lemma A.2.2 and Remark
+A.2.4 that (∂β)Ef is Pettis-integrable on Ω w.r.t. µβ for all β and that
+pα(∫
+Ω
+(∂β)Ef(x)dµβ(x)) ≤ ∣µβ∣(Kβ) sup
+x∈Kβ
+pα((∂β)Ef(x)),
+α ∈ A.
+By Example 3.2.8 a) the map S∶CW∞(Ω)εE → CW∞(Ω,E) is an isomorphism with
+inverse Rt∶f ↦ J ○ Rt
+f. The remaining parts of the proof follow from Proposition
+4.3.3 with (T E
+0 ,T K
+0 ) ∶= ((∂β)E,(∂β)K).
+□
+4.3.10. Proposition. Let E be a locally complete lcHs. Then for every T K ∈
+S(Rd)′ there is T E ∈ L(S(Rd,E),E) such that (T E,T K) is a consistent family for
+(S,E) and there are m ∈ N0 and a family of continuous functions (gβ)β∈Nd
+0,∣β∣≤m on
+Rd growing at infinity slower than some polynomial such that
+T E(f) = ∑
+∣β∣≤m
+∫
+Rd
+gβ(x)(∂β)Ef(x)dx,
+f ∈ S(Rd,E).
+(39)
+On the other hand, if (T E,T K) is a consistent family, then there are m ∈ N0 and a
+family of continuous functions (gβ)β∈Nd
+0,∣β∣≤m on Rd growing at infinity slower than
+some polynomial such that (39) holds and T E ∈ L(S(Rd,E),E).
+Proof. Let T K ∈ S(Rd)′. Then there are m ∈ N0 and a family of continuous
+functions (gβ)β∈Nd
+0,∣β∣≤m on Rd growing at infinity slower than some polynomial such
+that
+T K(f) = ∑
+∣β∣≤m
+∫
+Rd
+gβ(x)(∂β)Kf(x)dx,
+f ∈ S(Rd),
+by [171, Theorem 25.4, p. 272].
+Here, gβ growing at infinity slower than some
+polynomial means that there are k ∈ N0 and C > 0 such that ∣gβ(x)∣ ≤ C(1+∣x∣2)k/2
+for all x ∈ Rd. Since the family (gβ) is finite, we can take one k and one C for all
+β. Due to the proof of Example 3.2.9 and Corollary 3.2.10 we know that Kβ ∶=
+acx(((∂β)Ef)(1 + ∣ ⋅ ∣2)k/2(Rd)) is absolutely convex and compact for f ∈ S(Rd,E).
+The estimate
+∣∫
+Rd
+⟨e′,gβ(x)(∂β)Ef(x)⟩dx∣ ≤ C sup
+x∈Rd ∣e′((∂β)Ef(x))∣(1 + ∣x∣2)k/2 = C sup
+x∈Kβ
+∣e′(x)∣
+for all e′ ∈ E′ and f ∈ S(Rd,E) yields that gβ(∂β)Ef is Pettis-integrable on Rd
+w.r.t. the Lebesgue measure by the Mackey–Arens theorem. Further, it implies
+that
+pα(∫
+Rd
+gβ(x)(∂β)Ef(x)dx) ≤ C sup
+x∈Rd pα((∂β)Ef(x))(1 + ∣x∣2)k/2,
+α ∈ A,
+as in Lemma A.2.2. By Corollary 3.2.10 the map S∶S(Rd)εE → S(Rd,E) is an
+isomorphism with inverse Rt∶f ↦ J ○ Rt
+f. The remaining parts of the proof follow
+from Proposition 4.3.3 with (T E
+0 ,T K
+0 ) ∶= (gβ(∂β)E,gβ(∂β)K).
+□
+4.3.11. Remark.
+a) Let Ω ⊂ Rd be open and E a locally complete lcHs.
+Then Proposition 4.3.9 is still valid with CW∞ replaced by CW∞
+P (∂) due
+to the Hahn–Banach theorem and Example 3.2.8 b). If P(∂)K is a hy-
+poelliptic linear partial differential operator, then one can represent T E
+as in (37) due to Proposition 4.2.17 but the measure µ need not be unique
+anymore.
+
+66
+4. CONSISTENCY
+b) Let Ω ⊂ Rd be open and E an lcHs with metric ccp. Then Proposition
+4.3.7 is still valid with Cb replaced by C∞
+P (∂),b for a hypoelliptic linear
+partial differential operator P(∂)K due to the Hahn–Banach theorem and
+Proposition 4.2.24 but the measure µ need not be unique anymore.
+c) All families (T E,T K) considered in this section are strong which is a con-
+sequence of Proposition 4.3.4 (and of Pettis-summability in Proposition
+4.3.5).
+
+CHAPTER 5
+Applications
+5.1. Lifting the properties of maps from the scalar-valued case
+In this section we briefly show how to use the ε-compatibility of spaces F(Ω)
+and F(Ω,E) to lift properties like injectivity, surjectivity, bijectivity and continu-
+ity from a map T K to a map T E if (T E,T K) forms a consistent family. Especially,
+we pay attention to surjectivity whose transfer to the vector-valued case is ac-
+complished by Grothendieck’s classical theory of tensor products of Fréchet spaces
+[83] and by the splitting theory of Vogt for Fréchet spaces [173] and of Bonet and
+Domański for PLS-spaces [54]. In order to apply splitting theory, we recall the
+definitions of the topological invariants (Ω), (DN) and (PA).
+Let us recall that a Fréchet space F with an increasing fundamental system of
+seminorms (∣∣∣⋅∣∣∣k)k∈N satisfies (Ω) if
+∀ p ∈ N ∃ q ∈ N ∀ k ∈ N ∃ n ∈ N, C > 0 ∀ r > 0 ∶ Uq ⊂ CrnUk + 1
+r Up
+where Uk ∶= {x ∈ F ∣ ∣∣∣x∣∣∣k ≤ 1} (see [131, Chap. 29, Definition, p. 367]).
+We recall that a Fréchet space (F,(∣∣∣⋅∣∣∣k)k∈N) satisfies (DN) by [131, Chap. 29,
+Definition, p. 359] if
+∃ p ∈ N ∀ k ∈ N ∃ n ∈ N, C > 0 ∀ x ∈ F ∶ ∣∣∣x∣∣∣2
+k ≤ C∣∣∣x∣∣∣p∣∣∣x∣∣∣n.
+A PLS-space is a projective limit X = lim
+←�
+N∈N
+XN, where the XN given by inductive
+limits XN = lim
+�→
+n∈N
+(XN,n,∣∣∣⋅∣∣∣N,n) are DFS-spaces (which are also called LS-spaces),
+and it satisfies (PA) if
+∀ N ∃ M ∀ K ∃ n ∀ m ∀ η > 0 ∃ k,C,r0 > 0 ∀ r > r0 ∀ x′ ∈ X′
+N ∶
+∣∣∣x′ ○ iM
+N ∣∣∣
+∗
+M,m ≤ C(rη∣∣∣x′ ○ iK
+N∣∣∣
+∗
+K,k + 1
+r ∣∣∣x′∣∣∣
+∗
+N,n)
+where ∣∣∣⋅∣∣∣∗ denotes the dual norm of ∣∣∣⋅∣∣∣ and iM
+N , iK
+N the linking maps (see [26,
+Section 4, Eq. (24), p. 577]).
+Examples of Fréchet spaces with (DN) are the spaces of rapidly decreasing
+sequences s(Nd), s(Nd
+0) and s(Zd), the space C∞([a,b]) of all C∞-smooth functions
+on (a,b) such that all derivatives can be continuously extended to the boundary
+and the space of smooth functions C∞
+2π(Rd) that are 2π-periodic in each variable.
+Examples of ultrabornological PLS-space with (PA) are Fréchet–Schwartz spaces,
+the space of tempered distributions S(Rd)′
+b, the space of distributions D(Ω)′
+b and
+ultradistributions of Beurling type D(ω)(Ω)′
+b on an open set Ω ⊂ Rd. These and
+many more examples may be found in [26], [54, Corollary 4.8, p. 1116] and [112,
+Example 3, p. 7].
+5.1.1. Proposition.
+a) Let Y be a Fréchet space and X a semi-reflexive
+lcHs. Then Lb(X′
+b,Y ′
+b ) ≅ Lb(Y,(X′
+b)′
+b) via taking adjoints.
+b) Let E be an lcHs and X a Montel space. Then Lb(X′
+b,E) ≅ XεE where
+the isomorphism is the identity map.
+67
+
+68
+5. APPLICATIONS
+Proof. a) We consider the map
+t(⋅)∶Lb(X′
+b,Y ′
+b ) → Lb(Y,(X′
+b)′
+b), u ↦ tu,
+defined by tu(y)(x′) ∶= u(x′)(y) for y ∈ Y and x′ ∈ X′. First, we prove that t(⋅) is
+well-defined. Let u ∈ L(X′
+b,Y ′
+b ) and y ∈ Y . Since u ∈ L(X′
+b,Y ′
+b ) and {y} is bounded
+in Y , there are a bounded set B ⊂ X and C > 0 such that
+∣tu(y)(x′)∣ = ∣u(x′)(y)∣ ≤ C sup
+x∈B
+∣x′(x)∣
+for all x′ ∈ X′, implying tu(y) ∈ (X′
+b)′.
+Let us denote by (∥⋅∥Y,n)n∈N the (directed) system of seminorms generating the
+metrisable locally convex topology of Y . The canonical embedding J∶Y → (Y ′
+b )′
+b is
+an isomorphism between Y and J(Y ) by [131, Corollary 25.10, p. 298] because Y
+is a Fréchet space. For a bounded set M ⊂ X′
+b we note that
+sup
+x′∈M
+∣tu(y)(x′)∣ = sup
+x′∈M
+∣u(x′)(y)∣ = sup
+x′∈M
+∣⟨J(y),u(x′)⟩∣.
+The next step is to prove that u(M) is bounded in Y ′
+b . Let N ⊂ Y be bounded.
+Since u ∈ L(X′
+b,Y ′
+b ), there are again a bounded set B ⊂ X and a constant C > 0
+such that
+sup
+x′∈M
+sup
+y∈N
+∣u(x′)(y)∣ ≤ C sup
+x′∈M
+sup
+x∈B
+∣x′(x)∣ < ∞
+where the last estimate follows from the boundedness of M ⊂ X′
+b. Hence u(M) is
+bounded in Y ′
+b . By the remark about the canonical embedding there are n ∈ N and
+C0 > 0 such that
+sup
+x′∈M
+∣tu(y)(x′)∣ =
+sup
+y′∈u(M)
+∣⟨J(y),y′⟩∣ ≤ C0∥y∥Y,n,
+so tu ∈ L(Y,(X′
+b)′
+b) and the map t(⋅) is well-defined.
+Let us turn to injectivity. Let u,v ∈ L(X′
+b,Y ′
+b ) with tu = tv. This is equivalent
+to
+u(x′)(y) = tu(y)(x′) = tv(y)(x′) = v(x′)(y)
+for all y ∈ Y and x′ ∈ X′. This implies u(x′) = v(x′) for all x′ ∈ X′, hence u = v.
+Next, we turn to surjectivity. We consider the map
+t(⋅)∶Lb(Y,(X′
+b)′
+b) → Lb(X′
+b,Y ′
+b ), u ↦ tu,
+defined by tu(x′)(y) ∶= u(y)(x′) for x′ ∈ X′ and y ∈ Y . We show that this map is
+well-defined. Let u ∈ Lb(Y,(X′
+b)′
+b) and x′ ∈ X′. Since u ∈ Lb(Y,(X′
+b)′
+b) and {x′} is
+bounded in X′, there are n ∈ N and C > 0 such that
+∣tu(x′)(y)∣ = ∣u(y)(x′)∣ ≤ C∥y∥Y,n
+for all y ∈ Y , yielding tu(x′) ∈ Y ′. Let B ⊂ Y be bounded. The semi-reflexivity
+of X implies that for every u(y), y ∈ B, there is a unique xu(y) ∈ X such that
+u(y)(x′) = x′(xu(y)) for all x′ ∈ X′. Then we get
+sup
+y∈B
+∣tu(x′)(y)∣ = sup
+y∈B
+∣u(y)(x′)∣ = sup
+y∈B
+∣x′(xu(y))∣.
+We claim that D ∶= {xu(y) ∣ y ∈ B} is a bounded set in X. Let N ⊂ X′ be finite.
+Then the set M ∶= {tu(x′) ∣ x′ ∈ N} ⊂ Y ′ is finite. We have
+sup
+y∈B
+sup
+x′∈N
+∣x′(xu(y))∣ = sup
+y∈B
+sup
+x′∈N
+∣tu(x′)(y)∣ = sup
+y∈B
+sup
+y′∈M
+∣y′(y)∣ < ∞
+where the last estimate follows from the fact that the bounded set B is weakly
+bounded. Thus D is weakly bounded and by [131, Mackey’s theorem 23.15, p. 268]
+bounded in X. Therefore it follows from
+sup
+y∈B
+∣tu(x′)(y)∣ = sup
+y∈B
+∣x′(xu(y))∣ = sup
+x∈D
+∣x′(x)∣
+
+5.1. LIFTING THE PROPERTIES OF MAPS FROM THE SCALAR-VALUED CASE
+69
+for all x′ ∈ X′ that tu ∈ L(X′
+b,Y ′
+b ), which means that t(⋅) is well-defined.
+Let
+u ∈ L(Y,(X′
+b)′
+b). Then we have tu ∈ Lb(X′
+b,Y ′
+b ). In addition, for all y ∈ Y and all
+x′ ∈ X′
+t(tu)(y)(x′) = tu(x′)(y) = u(y)(x′)
+is valid and so t(tu)(y) = u(y) for all y ∈ Y , proving the surjectivity.
+The last step is to prove the continuity of t(⋅) and its inverse. Let M ⊂ Y and
+B ⊂ X′
+b be bounded sets. Then
+sup
+y∈M
+sup
+x′∈B
+∣tu(y)(x′)∣ = sup
+y∈M
+sup
+x′∈B
+∣u(x′)(y)∣ = sup
+x′∈B
+sup
+y∈M
+∣u(x′)(y)∣
+= sup
+x′∈B
+sup
+y∈M
+∣t(tu)(x′)(y)∣
+holds for all u ∈ L(X′
+b,Y ′
+b ). Therefore, t(⋅) and its inverse are continuous.
+b) Let T ∈ L(X′
+b,E). For α ∈ A there are a bounded set B ⊂ X and C > 0 such
+that
+pα(T(x′)) ≤ C sup
+x∈B
+∣x′(x)∣ ≤ C
+sup
+x∈acx(B)
+∣x′(x)∣
+for every x′ ∈ X′ where acx(B) is the closure of the absolutely convex hull of B.
+The set acx(B) is absolutely convex and compact by [89, 6.2.1 Proposition, p. 103]
+and [89, 6.7.1 Proposition, p. 112] since B is bounded in the Montel space X. Hence
+we gain T ∈ L(X′
+κ,E).
+Let M ⊂ X′ be equicontinuous. Due to [89, 8.5.1 Theorem (a), p. 156] M is
+bounded in X′
+b. Therefore,
+id∶Lb(X′
+b,E) → Le(X′
+κ,E) = XεE
+is continuous.
+Let T ∈ L(X′
+κ,E). For α ∈ A there are an absolutely convex compact set B ⊂ X
+and C > 0 such that
+pα(T(x′)) ≤ C sup
+x∈B
+∣x′(x)∣
+for every x′ ∈ X′. Since the compact set B is bounded, we get T ∈ L(X′
+b,E).
+Let M be a bounded set in X′
+b. Then M is equicontinuous by virtue of [171,
+Theorem 33.2, p. 349], as X, being a Montel space, is barrelled. Thus
+id∶Le(X′
+κ,E) → Lb(X′
+b,E)
+is continuous.
+□
+For part e) of the next theorem we need that our map SF2(Ω2)∶F2(Ω2)εE →
+F2(Ω2,E) is an isomorphism and that its inverse is given as in Theorem 3.2.4, i.e.
+that
+Rt∶F2(Ω2,E) → F2(Ω2)εE, f ↦ J −1 ○ Rt
+f,
+is the inverse of SF2(Ω2) where Rt
+f(f ′)(e′) = f ′(e′ ○ f) for f ′ ∈ F2(Ω2)′ and e′ ∈ E′,
+and J ∶E → E′⋆ is the canonical injection in the algebraic dual E′⋆ of E′.
+5.1.2. Theorem. Let E be an lcHs, F1(Ω1) and F1(Ω1,E) as well as F2(Ω2)
+and F2(Ω2,E) be ε-into-compatible. Let (T E,T K) be a consistent family for (F1,E)
+such that T K∶F1(Ω1) → F2(Ω2) is continuous and T E∶F1(Ω1,E) → F2(Ω2,E).
+Then the following holds:
+a) T E ○ SF1(Ω1) = SF2(Ω2) ○ (T KεidE).
+b) If SF1(Ω1) is surjective and T K is injective, then T E is injective, continu-
+ous and
+T E = SF2(Ω2) ○ (T KεidE) ○ S−1
+F1(Ω1).
+
+70
+5. APPLICATIONS
+If in addition SF2(Ω2) is surjective and T K an isomorphism, then T E is
+an isomorphism with inverse
+(T E)−1 = SF1(Ω1) ○ ((T K)−1εidE) ○ S−1
+F2(Ω2).
+c) If SF2(Ω2) and T KεidE are surjective, then T E is surjective.
+d) If SF2(Ω2) and T K are surjective, F1(Ω1), F2(Ω2) and E are Fréchet
+spaces and
+(i) F1(Ω1) and F2(Ω2) are nuclear, or
+(ii) E is nuclear,
+then T E is surjective.
+e) If SF2(Ω2) is surjective with inverse Rt, T K is surjective, F1(Ω1) and
+F2(Ω2) are Fréchet spaces, kerT K is nuclear and has (Ω), and
+(i) F1(Ω1) and F2(Ω2) are Montel spaces, E = F ′
+b where F is a Fréchet
+space satisfying (DN), or
+(ii) F1(Ω1) and F2(Ω2) are Schwartz spaces, E is an ultrabornological
+PLS-space satisfying (PA),
+then T E is surjective.
+Proof. a) Let u ∈ F1(Ω1)εE. Then
+(T E ○ SF1(Ω1))(u)(x) = u(δx ○ T K) = (u ○ (T K)t)(δx) = (T KεidE)(u)(δx)
+= SF2(Ω2)((T KεidE)(u))(x),
+x ∈ Ω2,
+as (T E,T K) is consistent for (F1,E), which proves part a).
+b) If SF1(Ω1) is surjective, then SF1(Ω1) is an isomorphism, because it is an
+isomorphism into, and we have
+T E = SF2(Ω2) ○ (T KεidE) ��� S−1
+F1(Ω1)
+by part a). If T K is injective, then T KεidE is also injective by [159, Chap. I, §1,
+Proposition 1, p. 20] and thus T E by the formula above as well since SF1(Ω1) is
+an isomorphism and SF2(Ω2) an isomorphism into. If SF2(Ω2) is surjective and T K
+an isomorphism, then SF2(Ω2) and T KεidE are isomorphisms, the latter by [159,
+Chap. I, §1, Proposition 1, p. 20] and its inverse is (T K)−1εidE. The rest of part
+b) follows from the formula for T E above.
+c) Let f ∈ F2(Ω2,E).
+Then there is g ∈ F1(Ω1)εE such that (SF2(Ω2) ○
+(T KεidE))(g) = f. Hence we obtain h ∶= SF1(Ω1)(g) ∈ F1(Ω1,E) and T E(h) = f by
+part a).
+d) For n = 1,2 the continuous linear injection (see (3))
+Θn∶Fn(Ωn) ⊗π E → Fn(Ωn)εE,
+k
+∑
+j=1
+fj ⊗ ej �→ [y ↦
+k
+∑
+j=1
+y(fj)ej],
+from the tensor product Fn(Ωn) ⊗π E with the projective topology extends to a con-
+tinuous linear map ̂Θn∶Fn(Ωn)̂⊗πE → Fn(Ωn)εE on the completion Fn(Ωn)̂⊗πE
+of Fn(Ωn) ⊗π E.
+The map ̂Θn is also a topological isomorphism since Fn(Ωn)
+is nuclear for n = 1,2 in case (i) resp. E is nuclear in case (ii).
+Furthermore,
+T K ⊗π idE∶F1(Ω1)⊗π E → F2(Ω2)⊗π E is defined by the relation Θ2 ○(T K ⊗π idE) =
+(T KεidE)○Θ1. We denote by T K ̂⊗π idE the continuous linear extension of T K⊗πidE
+to the completion F1(Ω1)̂⊗πE. Moreover, Fn(Ωn) for n = 1,2 and E are Fréchet
+spaces, T K and idE are linear, continuous and surjective, so T K ̂⊗π idE is surjective
+by [94, 10.24 Satz, p. 255]. We observe that
+T KεidE = ̂Θ2 ○ (T K ̂⊗π idE) ○ ̂Θ −1
+1
+
+5.1. LIFTING THE PROPERTIES OF MAPS FROM THE SCALAR-VALUED CASE
+71
+and deduce that T KεidE is surjective. Now, we apply part c), which proves part
+d).
+e) Throughout this proof we use the notation X′′ ∶= (X′
+b)′
+b for a locally convex
+Hausdorff space X and T ∶= T K. The space F1(Ω1) is a Fréchet space and so its
+closed subspace kerT as well. Further, Fn(Ωn) is a Montel space for n = 1,2 and
+kerT nuclear, thus they are reflexive. The sequence
+0 → kerT
+i→ F1(Ω1)
+T→ F2(Ω2) → 0,
+(40)
+where i means the inclusion, is a topologically exact sequence of Fréchet spaces
+because T is surjective by assumption. Let us denote by J0∶kerT → (kerT)′′ and
+Jn∶Fn(Ωn) → Fn(Ωn)′′ for n = 1,2 the canonical embeddings which are topological
+isomorphisms since kerT and Fn(Ωn) are reflexive for n = 1,2. Then the exactness
+of (40) implies that
+0 → (kerT)′′ i0→ F1(Ω1)′′ T1→ F2(Ω2)′′ → 0,
+(41)
+where i0 ∶= J0 ○ i ○ J−1
+0
+and T1 ∶= J2 ○ T ○ J−1
+1 , is an exact topological sequence. This
+exact sequence is topological because the (strong) bidual of a Fréchet space is again
+a Fréchet space by [131, Corollary 25.10, p. 298].
+(i) Let E = F ′
+b where F is a Fréchet space with (DN). Then Ext1(F,(kerT)′′) =
+0 by [174, 5.1 Theorem, p. 186] since kerT is nuclear and satisfies (Ω) and therefore
+(kerT)′′ as well. Combined with the exactness of (41) this implies that the sequence
+0 → L(F,(kerT)′′)
+i∗
+0→ L(F,F1(Ω1)′′)
+T ∗
+1→ L(F,F2(Ω2)′′) → 0
+is exact by [137, Proposition 2.1, p. 13–14] where i∗
+0(B) ∶= i0○B and T ∗
+1 (D) ∶= T1○D
+for B ∈ L(F,(kerT)′′) and D ∈ L(F,F1(Ω1)′′). In particular, we obtain that
+T ∗
+1 ∶L(F,F1(Ω1)′′) → L(F,F2(Ω2)′′)
+(42)
+is surjective. Via E = F ′
+b and Proposition 5.1.1 (X = Fn(Ωn) and Y = F) we have
+the isomorphisms into
+ψn ∶= SFn(Ωn) ○ t(⋅)∶L(F,Fn(Ωn)′′) → Fn(Ωn,E),
+ψn(u) = (SFn(Ωn) ○ t(⋅))(u) = [x ↦ tu(δx)],
+for n = 1,2 and the inverse
+ψ−1
+2 (f) = (S ○ t(⋅))−1(f) = (t(⋅) ○ S−1
+F2(Ω2))(f) = t(J −1 ○ Rt
+f),
+f ∈ F2(Ω2,E).
+Let g ∈ F2(Ω2,E). Then ψ−1
+2 (g) ∈ L(F,F2(Ω2)′′) and by the surjectivity of (42)
+there is u ∈ L(F,F1(Ω1)′′) such that T ∗
+1 u = ψ−1
+2 (g). So we get ψ1(u) ∈ F1(Ω1,E).
+Next, we show that T Eψ1(u) = g is valid. Let y ∈ F and x ∈ Ω2. Then
+T E(ψ1(u))(x) = tu(δx ○ T)
+by consistency and
+T E(ψ1(u))(x)(y) = tu(δx ○ T)(y) = u(y)(δx ○ T) = ⟨δx ○ T,J−1
+1 (u(y))⟩
+= ⟨δx,TJ−1
+1 (u(y))⟩ = ⟨[J2 ○ T ○ J−1
+1 ](u(y)),δx⟩ = ⟨(T1 ○ u)(y),δx⟩
+= ⟨(T ∗
+1 u)(y),δx⟩ = ψ−1
+2 (g)(y)(δx) = t(J −1 ○ Rt
+g)(y)(δx)
+= (J −1 ○ Rt
+g)(δx)(y) = J −1(J (g(x))(y) = g(x)(y).
+Thus T E(ψ1(u))(x) = g(x) for every x ∈ Ω2, which proves the surjectivity.
+(ii) Let E be an ultrabornological PLS-space satisfying (PA). Since the nuclear
+Fréchet space kerT is also a Schwartz space, its strong dual (kerT)′
+b is a DFS-
+space.
+By [26, Theorem 4.1, p. 577] we obtain Ext1
+P LS((kerT)′
+b,E) = 0 as the
+bidual (kerT)′′ satisfies (Ω), E is a PLS-space satisfying (PA) and condition (c)
+in the theorem is fulfilled because (kerT)′
+b is the strong dual of a nuclear Fréchet
+
+72
+5. APPLICATIONS
+space. Moreover, we have Proj1 E = 0 due to [180, Corollary 3.3.10, p. 46] because
+E is an ultrabornological PLS-space.
+Then the exactness of the sequence (41),
+[26, Theorem 3.4, p. 567] and [26, Lemma 3.3, p. 567] (in the lemma the same
+condition (c) as in [26, Theorem 4.1, p. 577] is fulfilled and we choose H = (kerT)′′,
+F = F1(Ω1)′′ and G = F2(Ω2)′′), imply that the sequence
+0 → L(E′
+b,(kerT)′′)
+i∗
+0→ L(E′
+b,F1(Ω1)′′)
+T ∗
+1→ L(E′
+b,F2(Ω2)′′) → 0
+is exact. The maps i∗
+0 and T ∗
+1 are defined as in part (i). Especially, we get that
+T ∗
+1 ∶L(E′
+b,F1(Ω1)′′) → L(E′
+b,F2(Ω2)′′)
+(43)
+is surjective.
+By [54, Remark 4.4, p. 1114] we have Lb(Fn(Ωn)′
+b,E′′) ≅ Lb(E′
+b,Fn(Ωn)′′) for
+n = 1,2 via taking adjoints since Fn(Ωn), being a Fréchet–Schwartz space, is a PLS-
+space and hence its strong dual an LFS-space, which is regular by [180, Corollary
+6.7, 10. ⇔ 11., p. 114], and E is an ultrabornological PLS-space, in particular,
+reflexive by [53, Theorem 3.2, p. 58]. In addition, the map
+P∶Lb(Fn(Ωn)′
+b,E′′) → Lb(Fn(Ωn)′
+b,E),
+defined by P(u)(y) ∶= J −1(u(y)) for u ∈ L(Fn(Ωn)′
+b,E′′) and y ∈ Fn(Ωn)′, is an
+isomorphism because E is reflexive. Due to Proposition 5.1.1 b) with X = Fn(Ωn)
+we obtain the isomorphisms into
+ψn ∶= S ○ J −1 ○ t(⋅)∶Lb(E′
+b,Fn(Ωn)′′) → Fn(Ωn,E),
+ψn(u) = [SFn(Ωn) ○ J −1 ○ t(⋅)](u) = [x ↦ J −1(tu(δx))],
+for n = 1,2 and the inverse given by
+ψ−1
+2 (f) = (SF2(Ω2) ○ J −1 ○ t(⋅))−1(f) = [t(⋅) ○ J ○ S−1
+F2(Ω2)](f) = t(J ○ J −1 ○ Rt
+f)
+= t(Rt
+f)
+for f ∈ F2(Ω2,E).
+Let g ∈ F2(Ω2,E).
+Then ψ−1
+2 (g) ∈ L(E′
+b,F2(Ω2)′′) and by the surjectivity
+of (43) there exists u ∈ L(E′
+b,F1(Ω1)′′) such that T ∗
+1 u = ψ−1
+2 (g).
+So we have
+ψ1(u) ∈ F1(Ω1,E). The last step is to show that T Eψ1(u) = g. As in part (i) we
+gain for every x ∈ Ω2
+T E(ψ1(u))(x) = J −1(tu(δx ○ T))
+by consistency and for every y ∈ E′
+tu(δx ○ T)(y) = u(y)(δx ○ T) = (T ∗
+1 u)(y)(δx) = ψ−1
+2 (g)(y)(δx) = t(Rt
+g)(y)(δx)
+= δx(y ○ g) = y(g(x)) = J (g(x))(y).
+Thus we have tu(δx ○ T) = J (g(x)) and therefore T E(ψ1(u))(x) = g(x) for all
+x ∈ Ω2.
+□
+Theorem 5.1.2 d) and e) are generalisations of [116, Corollary 4.3, p. 2689] and
+[112, Theorem 5, p. 7–8] where T C is the Cauchy–Riemann operator ∂ on certain
+weighted spaces CV∞(Ω) of smooth functions. Our next result is the well-known
+application of tensor product theory and splitting theory to linear partial differential
+operators we already mentioned in the introduction.
+5.1.3. Corollary. Let E be a locally complete lcHs, Ω1 ⊂ Rd open and P(∂)K
+be a linear partial differential operator with C∞-smooth coefficients. Then the fol-
+lowing holds:
+a) P(∂)E = SC∞(Ω1) ○ (P(∂)KεidE) ○ S−1
+C∞(Ω1).
+b) If K = C, P(D) ∶= P(D)C ∶= P(−i∂)C has constant coefficients and is
+
+5.1. LIFTING THE PROPERTIES OF MAPS FROM THE SCALAR-VALUED CASE
+73
+(i) elliptic, or
+(ii) hypoelliptic and Ω1 convex,
+and
+(iii) E is a Fréchet space, or
+(iv) E = F ′
+b where F is a Fréchet space satisfying (DN), d ≥ 2, or
+(v) E is an ultrabornological PLS-space satisfying (PA), d ≥ 2,
+then P(D)E∶C∞(Ω1,E) → C∞(Ω1,E) is surjective.
+Proof. Part a) follows from Theorem 5.1.2 a), Example 3.2.8 a) and the con-
+sistency of (P(∂)E,P(∂)K) because (7) holds for u ∈ CW∞(Ω1)εE as well.
+Let us turn to part b). The inverse of SC∞(Ω1) is given by Rt by Example 3.2.8
+a). The map P(D) = P(D)C∶C∞(Ω1) → C∞(Ω1) is surjective by [86, Corollary
+10.6.8, p. 43] and [86, Theorem 10.6.2, p. 41] in case (ii) resp. by [86, Corollary
+10.8.2, p. 51] in case (i). The space CW∞(Ω1), i.e. C∞(Ω1) with its usual topology
+(see Example 3.1.9 b)), is a nuclear Fréchet space and thus its closed subspace
+kerP(D)C as well. In case (i) kerP(D)C has (Ω) due to [173, Proposition 2.5 (b),
+p. 173] and in case (ii) due to [140, 4.5 Corollary (a), p. 202]. Hence the surjectivity
+of P(D)E follows from Theorem 5.1.2 d)+e).
+□
+Recently, it was shown in [47, Theorem 4.2, p. 13] that in the case that
+Ω1 is convex, kerP(D)C has property (Ω) for any P(D) with constant coeffi-
+cients (so without the assumption of P(D) being hypoelliptic).
+Hence we may
+replace the assumption of P(D) being hypoelliptic in (ii) by the assumption that
+P(D)C∶C∞(Ω1) → C∞(Ω1) is surjective. Even more recently, a necessary and suf-
+ficient condition for the surjectivity of P(D)C and kerP(D)C having property (Ω)
+for P(D) with constant coefficients and general open Ω1 ⊂ Rd was derived in [48,
+Theorem 1.1. (a), p. 3] using shifted fundamental solutions.
+Even though Corollary 5.1.3 b) is known, it is often proved without using tensor
+products or splitting theory (see e.g. [94, Theorem 10.10, p. 240]) or it is phrased
+as the surjectivity of P(D)Ĉ⊗π idE (see e.g. [171, Eq. (52.4), p. 541]) and the proof
+of the relation
+P(D)E = SC∞(Ω1) ○ (̂Θ1 ○ (P(D)C ̂⊗π idE) ○ ̂Θ −1
+1 ) ○ S−1
+C∞(Ω1)
+for Fréchet spaces E is omitted (see e.g. [171, p. 545–546]), or only the surjectivity
+of T ∗
+1 = P(D)∗
+1 in part e) of Theorem 5.1.2 is actually shown and it is only stated
+but not proved that this implies the surjectivity of P(D)E (see e.g. the statement
+of surjectivity of P(D)E in [173, p. 168] for elliptic P(D) and E = F ′
+b for a Fréchet
+space F with (DN) and that it is ‘only’ shown that P(D)∗
+1 is surjective by [173,
+Proposition 2.5 (b), p. 173] and [173, Theorem 2.4 (b), p. 173] where the symbol
+P(D)∗ is used instead of P(D)∗
+1 in [173, p. 172] since the isomorphism J1 = J2 is
+omitted). So, apart from being the probably most classical application of tensor
+products or splitting theory, that is the reason why we still included Corollary 5.1.3.
+Let us give another application of Theorem 5.1.2 d) and e), namely, a vector-
+valued Borel–Ritt theorem.
+5.1.4. Theorem. Let E be an lcHs and (xn)n∈N0 a sequence in E. If
+(i) E is a Fréchet space, or
+(ii) E = F ′
+b where F is a Fréchet space satisfying (DN), or
+(iii) E is an ultrabornological PLS-space satisfying (PA),
+then there is f ∈ C∞
+2π(R,E) such that (∂n)Ef(0) = xn for all n ∈ N0.
+Proof. By the Borel–Ritt theorem [94, Satz 9.12, p. 206] the map
+T K∶C∞
+2π(R) → KN0, T K(f) ∶= ((∂n)Kf(0))n∈N0,
+
+74
+5. APPLICATIONS
+is surjective and obviously linear and continuous as well. Now, we define the map
+T E∶C∞
+2π(R,E) → EN0 by replacing K by E in the definition of T K. Due to Example
+4.2.1 KN0 and EN0 are ε-compatible and the inverse of SKN0 is given by Rt. In
+addition, C∞
+2π(R) and C∞
+2π(R,E) are ε-compatible by Example 4.2.27 as in all three
+cases E is complete. We observe that (T E,T K) is consistent by Proposition 3.1.11
+c).
+The spaces KN0 and C∞
+2π(R) are nuclear Fréchet spaces.
+The first by [171,
+Theorem 51.1, p. 526] and the second because it is a subspace of the nuclear space
+C∞(R) by [131, Examples 28.9 (1), p. 349–350] and [131, Proposition 28.6, p. 347].
+Hence in case (i) our statement follows from Theorem 5.1.2 d). Moreover, kerT K
+is nuclear since C∞
+2π(R) is nuclear. By the proof of [131, Lemma 31.3, p. 392–393]
+kerT K is isomorphic to s(N0). The space s(N0) has (Ω) by [131, Lemma 29.11
+(3), p. 368] and thus kerT K as well because (Ω) is a linear topological invariant
+by [131, Lemma 29.11 (1), p. 368]. Therefore our statement in case (ii) and (iii)
+follows from Theorem 5.1.2 e).
+□
+We close this section with an application of Theorem 5.1.2 b) to the Fourier
+transformation on the Beurling–Björck spaces Sµ(Rd,E) from Example 4.2.26.
+5.1.5. Theorem. Let E be a locally complete lcHs over C and µ∶Rd → [0,∞)
+continuous such that µ(x) = µ(−x) for all x ∈ Rd and condition (γ) is fulfilled.
+(i) If E has metric ccp, or
+(ii) if µ ∈ C1(Rd) and there are k ∈ N0, C > 0 such that ∣∂enµ(x)∣ ≤ Cekµ(x)
+for all x ∈ Rd and 1 ≤ n ≤ d,
+then FE∶Sµ(Rd,E) → Sµ(Rd,E) is an isomorphism with FE = S ○ (FCεidE) ○ S−1.
+Proof. Due to Example 4.2.26 Sµ(Rd) and Sµ(Rd,E) are ε-compatible. The
+Fourier transformation FC∶Sµ(Rd) → Sµ(Rd) is a well-defined isomorphism by the
+definition of Sµ(Rd) and since (FC ○ FC)(f)(x) = f(−x) for all f ∈ Sµ(Rd) as well
+as µ(x) = µ(−x) for all x ∈ Rd. Due to (26) with β = 0 we have that (FE,FC) is a
+consistent family for (Sµ,E) and thus it follows from Theorem 5.1.2 b) that FE is
+an isomorphism and FE = S ○ (FCεidE) ○ S−1, which completes the proof.
+□
+5.2. Extension of vector-valued functions
+We study the problem of extending vector-valued functions via the existence
+of weak extensions in this section. The precise description of this problem reads
+as follows. Let E be a locally convex Hausdorff space over the field K of real or
+complex numbers and F(Ω) ∶= F(Ω,K) a locally convex Hausdorff space of K-
+valued functions on a set Ω. Suppose that the point evaluations δx belong to the
+dual F(Ω)′ for every x ∈ Ω and that there is a locally convex Hausdorff space
+F(Ω,E) of E-valued functions on Ω such that the map
+S∶F(Ω)εE → F(Ω,E), u �→ [x ↦ u(δx)],
+(44)
+is an isomorphism into, i.e. F(Ω) and F(Ω,E) are ε-into-compatible. Thus F(Ω)εE
+is a linearisation of a subspace of F(Ω,E). Linearisations that are based on the
+Dixmier–Ng theorem were used by Bonet, Domański and Lindström in [28, Lemma
+10, p. 243] resp. Laitila and Tylli in [121, Lemma 5.2, p. 14] to describe the space
+of weakly holomorphic resp. harmonic functions on the unit disc Ω = D ⊂ C with
+values in a (complex) Banach space E (see also [118]).
+5.2.1. Question. Let Λ be a subset of Ω and G a linear subspace of E′. Let
+f∶Λ → E be such that for every e′ ∈ G, the function e′ ○ f∶Λ → K has an extension
+in F(Ω). When is there an extension F ∈ F(Ω,E) of f, i.e. F∣Λ = f ?
+
+5.2. EXTENSION OF VECTOR-VALUED FUNCTIONS
+75
+An affirmative answer for Λ = Ω and G = E′ is called a weak-strong principle.
+For weighted continuous functions on a completely regular Hausdorff space Ω with
+values in a semi-Montel or Schwartz space E a weak-strong principle is given by
+Bierstedt in [17, 2.10 Lemma, p. 140].
+Weak-strong principles for holomorphic
+functions on open subsets Ω ⊂ C were shown by Dunford in [59, Theorem 76, p.
+354] for Banach spaces E and by Grothendieck in [82, Théorème 1, p. 37–38] for
+quasi-complete E. For a wider class of function spaces weak-strong principles are
+due to Grothendieck, mainly, in the case that F(Ω) is nuclear and E complete (see
+[83, Chap. II, §3, n○3, Théorème 13, p. 80]), which covers the case that F(Ω) is the
+space C∞(Ω) of smooth functions on an open set Ω ⊂ Rd (with its usual topology).
+Gramsch [77] analysed the weak-strong principles of Grothendieck and realised
+that they can be used to extend functions if Λ is a set of uniqueness, i.e. from
+f ∈ F(Ω) and f(x) = 0 for all x ∈ Λ follows that f = 0, and F(Ω) a semi-Montel
+space, E complete and G = E′ (see [77, 0.1, p. 217]).
+An extension result for
+holomorphic functions where G = E′ and E is sequentially complete was shown by
+Bogdanowicz in [25, Corollary 3, p. 665].
+Grosse-Erdmann proved for holomorphic functions on Λ = Ω in [79, 5.2 The-
+orem, p. 35] that it is sufficient to test locally bounded functions f with values
+in a locally complete space E with functionals from a weak⋆-dense subspace G of
+E′. Arendt and Nikolski [7, 8] shortened his proof in the case that E is a Fréchet
+space (see [7, Theorem 3.1, p. 787] and [7, Remark 3.3, p. 787]). Arendt gave an
+affirmative answer in [6, Theorem 5.4, p. 74] for harmonic functions on an open
+subset Λ = Ω ⊂ Rd where the range space E is a Banach space and G a weak⋆-dense
+subspace of E′.
+In [77] Gramsch also derived extension results for a large class of Fréchet–
+Montel spaces F(Ω) in the case that Λ is a special set of uniqueness, E sequentially
+complete and G strongly dense in E′ (see [77, 3.3 Satz, p. 228–229]). He applied
+it to the space of holomorphic functions and Grosse-Erdmann [81] expanded this
+result to the case of E being Br-complete and G only a weak⋆-dense subspace of
+E′ (see [81, Theorem 2, p. 401] and [81, Remark 2 (a), p. 406]). In a series of
+papers [30, 69, 70, 92, 93] these results were generalised and improved by Bonet,
+Frerick, Jordá and Wengenroth who used (44) to obtain extensions for vector-valued
+functions via extensions of linear operators. In [92, 93] this was done by Jordá for
+holomorphic functions on a domain (i.e. open and connected) Ω ⊂ C and weighted
+holomorphic functions on a domain Ω in a Banach space. In [30] this was done
+by Bonet, Frerick and Jordá for closed subsheaves F(Ω) of the sheaf of smooth
+functions C∞(Ω) on a domain Ω ⊂ Rd. Their results implied some consequences
+on the work of Bierstedt and Holtmanns [18] as well. Further, in [69] this was
+done by Frerick and Jordá for closed subsheaves F(Ω) of smooth functions on a
+domain Ω ⊂ Rd which are closed in the sheaf C(Ω) of continuous functions and in
+[70] by the first two authors and Wengenroth in the case that F(Ω) is the space of
+bounded functions in the kernel of a hypoelliptic linear partial differential operator,
+in particular, the spaces of bounded holomorphic or harmonic functions.
+In this section we present a unified approach to the extension problem for a large
+class of function spaces. The spaces we treat are usually of the kind that F(Ω)
+belongs to the class of semi-Montel spaces, Fréchet–Schwartz spaces or Banach
+spaces.
+Even quite general weighted spaces F(Ω) are treated, at least, if E is
+a semi-Montel space.
+Our approach is based on three ideas.
+First, it is based
+on the representation of (a subspace of) F(Ω,E) as a space of continuous linear
+operators via the map S from (44). We note that almost all our examples of such
+spaces F(Ω,E) are actually of the form of a general weighted space FV(Ω,E) from
+Definition 3.1.4. Second, it is based on the idea to consider a set of uniqueness Λ
+
+76
+5. APPLICATIONS
+not necessarily as a subset of Ω but rather as a set of functionals acting on F(Ω).
+In the definition of a set of uniqueness given above one may identify Λ with the set
+of functionals {δx ∣ x ∈ Λ} and this shift of perspective allows us to consider certain
+sets of functionals of the form T K
+m,x as sets of uniqueness for F(Ω) (see Definition
+5.2.2). Third, the generalised concept of consistency and strength of a family of
+operators (T E
+m,T K
+m)m∈M acting on (F(Ω,E),F(Ω)) from Definition 4.3.1 enables
+us to generalise Question 5.2.1 and affirmatively answer this generalised question.
+These three ideas are used to extend the mentioned results and we always have
+to balance the sets Λ from which we extend our functions and the subspaces G ⊂ E′
+with which we test. The case of ‘thin’ sets Λ and ‘thick’ subspaces G is handled in
+Section 5.2.1, the converse case of ‘thick’ sets Λ and ‘thin’ subspaces G in Section
+5.2.2.
+5.2.1. Extension from thin sets. Using the functionals T K
+m,x, we extend the
+definition of a set of uniqueness and a space of restrictions given in [30, Definition
+4, 5, p. 230]. This prepares the ground for a generalisation of Question 5.2.1 using
+a strong, consistent family (T E
+m,T K
+m)m∈M.
+5.2.2. Definition (set of uniqueness). Let Ω be a non-empty set, F(Ω) ⊂ KΩ
+an lcHs, (ωm)m∈M be a family of non-empty sets and T K
+m∶F(Ω) → Kωm be linear for
+all m ∈ M. Then U ⊂ ⋃m∈M({m}×ωm) is called a set of uniqueness for (T K
+m,F)m∈M
+if
+(i) ∀ (m,x) ∈ U ∶ T K
+m,x ∈ F(Ω)′,
+(ii) ∀ f ∈ F(Ω) ∶ [∀(m,x) ∈ U ∶ T K
+m(f)(x) = 0] ⇒ f = 0.
+We omit the index m in ωm and T K
+m if M is a singleton and consider U as a subset
+of ω.
+If U is a set of uniqueness for (T K
+m,F)m∈M, then span{T K
+m,x ∣ (m,x) ∈ U} is
+dense in F(Ω)′
+σ (and F(Ω)′
+κ) by the bipolar theorem.
+5.2.3. Remark. Let Ω be a non-empty set and F(Ω) ⊂ KΩ an lcHs.
+a) A simple set of uniqueness for (idKΩ,F) is given by U ∶= Ω if δx ∈ F(Ω)′
+for all x ∈ Ω.
+b) If F(Ω) has a Schauder basis (fn)n∈N with associated sequence of coef-
+ficient functionals T K ∶= (T K
+n )n∈N, then U ∶= N is a set of uniqueness for
+(T K,F).
+An example for b) is the space of holomorphic functions on an open disc
+Dr(z0) ⊂ C with radius 0 < r ≤ ∞ and center z0 ∈ C.
+If we equip this space
+with the topology of compact convergence, then it has the shifted monomials
+((⋅ − z0)n)n∈N0 as a Schauder basis with the point evaluations (δz0 ○ ∂n
+C)n∈N0 given
+by (δz0 ○ ∂n
+C)(f) ∶= f (n)(z0) as associated sequence of coefficient functionals. We
+will explore further sets of uniqueness for concrete function spaces in the upcoming
+examples and come back to b) in Section 5.7.
+5.2.4. Definition (restriction space). Let G ⊂ E′ be a separating subspace
+and U a set of uniqueness for (T K
+m,F)m∈M. Let FG(U,E) be the space of functions
+f∶U → E such that for every e′ ∈ G there is fe′ ∈ F(Ω) with T K
+m(fe′)(x) = (e′ ○
+f)(m,x) for all (m,x) ∈ U.
+5.2.5. Remark. Since U is a set of uniqueness, the functions fe′ are unique
+and the map Rf∶E′ → F(Ω), Rf(e′) ∶= fe′, is well-defined and linear. The map Rf
+resembles the map Rf defined above Lemma 3.2.1.
+5.2.6. Remark. Let F(Ω) and F(Ω,E) be ε-into-compatible. Consider a set
+of uniqueness U for (T K
+m,F)m∈M, a separating subspace G ⊂ E′ and a strong,
+
+5.2. EXTENSION OF VECTOR-VALUED FUNCTIONS
+77
+consistent family (T E
+m,T K
+m)m∈M for (F,E). For u ∈ F(Ω)εE set f ∶= S(u). Then
+f ∈ F(Ω,E) by the ε-into-compatibility and we set ̃f∶U → E, ̃f(m,x) ∶= T E
+m(f)(x).
+It follows that
+(e′ ○ ̃f)(m,x) = (e′ ○ T E
+m(f))(x) = T K
+m(e′ ○ f)(x)
+for all (m,x) ∈ U and fe′ ∶= e′ ○f ∈ F(Ω) for all e′ ∈ E′ by the strength of the family.
+We conclude that ̃f ∈ FG(U,E).
+5.2.7. Remark. If U is a set of uniqueness for (T K
+m,F)m∈M, then the existence
+of operators (T E
+m)m∈M such that (T E
+m,T K
+m)m∈M is a strong, consistent family for
+(F,E) is often guaranteed by the Riesz–Markov–Kakutani representation theorems
+in Section 4.3.
+Under the assumptions of Remark 5.2.6 the map
+RU,G∶S(F(Ω)εE) → FG(U,E), f ↦ (T E
+m(f)(x))(m,x)∈U,
+is well-defined.
+The map RU,G is also linear since T E
+m is linear for all m ∈ M.
+Further, the strength of the defining family guarantees that RU,G is injective.
+5.2.8. Proposition. Let F(Ω) and F(Ω,E) be ε-into-compatible, G ⊂ E′ a
+separating subspace and U a set of uniqueness for (T K
+m,F)m∈M. If (T E
+m,T K
+m)m∈M is
+a strong family for (F,E), then the map
+T E∶F(Ω,E) → EU, f ↦ (T E
+m(f)(x))(m,x)∈U,
+is injective, in particular, RU,G is injective.
+Proof. Let f ∈ F(Ω,E) with T E(f) = 0. Then
+0 = (e′ ○ T E(f))(m,x) = (e′ ○ T E
+m(f))(x) = T K
+m(e′ ○ f)(x),
+(m,x) ∈ U,
+and e′ ○ f ∈ F(Ω) for all e′ ∈ E′ by the strength of the family. Since U is a set of
+uniqueness, we get that e′ ○ f = 0 for all e′ ∈ E′, which implies f = 0.
+□
+5.2.9. Question. Let F(Ω) and F(Ω,E) be ε-into-compatible, G ⊂ E′ a sepa-
+rating subspace, (T E
+m,T K
+m)m∈M a strong family for (F,E) and U a set of uniqueness
+for (T K
+m,F)m∈M. When is the injective restriction map
+RU,G∶S(F(Ω)εE) → FG(U,E), f ↦ (T E
+m(f)(x))(m,x)∈U,
+surjective?
+The Question 5.2.1 is a special case of this question if there is a set of uniqueness
+U for (T K
+m,F)m∈M with {T K
+m,x ∣ (m,x) ∈ U} = {δx ∣ x ∈ Λ}, Λ ⊂ Ω. We observe that
+a positive answer to the surjectivity of RΩ,G results in the following weak-strong
+principle.
+5.2.10. Proposition. Let F(Ω) and F(Ω,E) be ε-into-compatible, G ⊂ E′ a
+separating subspace such that e′ ○ f ∈ F(Ω) for all e′ ∈ G and f ∈ F(Ω,E). If
+RΩ,G∶S(F(Ω)εE) → FG(Ω,E), f ↦ f,
+with the set of uniqueness Ω for (idKΩ,F) is surjective, then
+F(Ω)εE ≅ F(Ω,E)
+via S
+and
+F(Ω,E) = {f∶Ω → E ∣ ∀ e′ ∈ G ∶ e′ ○ f ∈ F(Ω)}.
+Proof. From the ε-into-compatibility and the surjectivity of RΩ,G we obtain
+{f∶Ω → E ∣ ∀ e′ ∈ G ∶ e′ ○ f ∈ F(Ω)} = FG(Ω,E) = S(F(Ω)εE) ⊂ F(Ω,E).
+Further, the assumption that e′ ○ f ∈ F(Ω) for all e′ ∈ G and f ∈ F(Ω,E), implies
+that F(Ω,E) is a subspace of the space on the left-hand side, which proves our
+statement, in particular, the surjectivity of S.
+□
+
+78
+5. APPLICATIONS
+To answer Question 5.2.9 for general sets of uniqueness we have to restrict to
+a certain class of separating subspaces of E′.
+5.2.11. Definition (determine boundedness [30, p. 230]). A linear subspace
+G ⊂ E′ determines boundedness if every σ(E,G)-bounded set B ⊂ E is already
+bounded in E.
+In [67, p. 139] such a space G is called uniform boundedness deciding by Fer-
+nández et al. and in [134, p. 63] w∗-thick by Nygaard if E is a Banach space.
+5.2.12. Remark.
+a) Let E be an lcHs. Then G ∶= E′ determines bound-
+edness by [131, Mackey’s theorem 23.15, p. 268].
+b) Let X be a barrelled lcHs, Y an lcHs and E ∶= Lb(X,Y ). For x ∈ X and
+y′ ∈ Y ′ we set δx,y′∶L(X,Y ) → K, T ↦ y′(T(x)), and G ∶= {δx,y′ ∣ x ∈
+X, y′ ∈ Y ′} ⊂ E′. Then the span of G determines boundedness (in E) by
+Mackey’s theorem and the uniform boundedness principle. For Banach
+spaces X,Y this is already observed in [30, Remark 11, p. 233] and, if in
+addition Y = K, in [7, Remark 1.4 b), p. 781].
+c) Further examples and a characterisation of subspaces G ⊂ E′ that de-
+termine boundedness can be found in [7, Remark 1.4, p. 781–782], [134,
+Theorem 1.5, p. 63–64] and [134, Theorem 2.3, 2.4, p. 67–68] in the case
+that E is a Banach space.
+F(Ω) a semi-Montel space and E (sequentially) complete. Our next
+results are in need of spaces F(Ω) such that closed graph theorems hold with
+Banach spaces as domain spaces and F(Ω) as the range space. Let us formally
+define this class of spaces.
+5.2.13. Definition (BC-space [142, p. 395]). We call an lcHs F a BC-space
+if for every Banach space X and every linear map f∶X → F with closed graph in
+X × F, one has that f is continuous.
+A characterisation of BC-spaces is given by Powell in [142, 6.1 Corollary, p. 400–
+401]. Since every Banach space is ultrabornological and barrelled, the [131, Closed
+graph theorem 24.31, p. 289] of de Wilde and the Pták–K¯omura–Adasch–Valdivia
+closed graph theorem [103, §34, 9.(7), p. 46] imply that webbed spaces and Br-
+complete spaces are BC-spaces. We recall that an lcHs F is said to be Br-complete
+if every σ(F ′,F)-dense σf(F ′,F)-closed linear subspace of F ′ equals F ′ where
+σf(F ′,F) is the finest topology coinciding with σ(F ′,F) on all equicontinuous sets
+in F ′ (see [103, §34, p. 26]). An lcHs F is called B-complete if every σf(F ′,F)-closed
+linear subspace of F ′ is weakly closed. In particular, B-complete spaces are Br-
+complete and every Br-complete space is complete by [103, §34, 2.(1), p. 26]. These
+definitions are equivalent to the original definitions of Br- and B-completeness by
+Pták [143, Definition 2, 5, p. 50, 55] due to [103, §34, 2.(2), p. 26–27] and we
+note that they are also called infra-Pták spaces and Pták spaces, respectively. In
+particular, Fréchet spaces are B-complete by [89, 9.5.2 Krein–˘Smulian Theorem, p.
+184] but we will encounter non-Fréchet B-complete spaces as well.
+The following proposition is a modification of [94, Satz 10.6, p. 237] and uses
+the map Rf∶e′ ↦ fe′ from Remark 5.2.5.
+5.2.14. Proposition. Let U be a set of uniqueness for (T K
+m,F)m∈M and F(Ω)
+a BC-space. Then Rf(B○
+α) is bounded in F(Ω) for every f ∈ FE′(U,E) and α ∈ A
+where Bα ∶= {x ∈ E ∣ pα(x) < 1}. In addition, if F(Ω) is a semi-Montel space, then
+Rf(B○
+α) is relatively compact in F(Ω).
+
+5.2. EXTENSION OF VECTOR-VALUED FUNCTIONS
+79
+Proof. Let f ∈ FE′(U,E) and α ∈ A. The polar B○
+α is compact in E′
+σ and
+thus E′
+B○α is a Banach space by [131, Corollary 23.14, p. 268].
+We claim that
+the restriction of Rf to E′
+B○α has closed graph. Indeed, let (e′
+τ) be a net in E′
+B○α
+converging to e′ in E′
+B○α and Rf(e′
+τ) converging to g in F(Ω). For (m,x) ∈ U we
+note that
+T K
+m,x(Rf(e′
+τ)) = T K
+m(fe′τ )(x) = (e′
+τ ○ f)(m,x) → (e′ ○ f)(m,x) = T K
+m(fe′)(x)
+= T K
+m(Rf(e′))(x).
+The left-hand side converges to T K
+m,x(g) since T K
+m,x ∈ F(Ω)′ for all (m,x) ∈ U.
+Hence we have T K
+m(g)(x) = T K
+m(Rf(e′))(x) for all (m,x) ∈ U. From U being a
+set of uniqueness follows that g = Rf(e′). Thus the restriction of Rf to E′
+B○α has
+closed graph and is continuous since F(Ω) is a BC-space. This yields that Rf(B○
+α)
+is bounded as B○
+α is bounded in E′
+B○α. If F(Ω) is also a semi-Montel space, then
+Rf(B○
+α) is even relatively compact.
+□
+Now, we are ready to prove our first extension theorem. Its proof of surjectivity
+of RU,E′ is just an adaptation of the proof of surjectivity of S given in Theorem
+3.2.4. Let U be a set of uniqueness for (T K
+m,F)m∈M. For f ∈ FE′(U,E) we consider
+the dual map
+Rt
+f∶F(Ω)′ → E′⋆, Rt
+f(y)(e′) ∶= y(fe′),
+where E′⋆ is the algebraic dual of E′. Further, we recall the notation J ∶E → E′⋆
+for the canonical injection.
+5.2.15. Theorem. Let F(Ω) and F(Ω,E) be ε-into-compatible, (T E
+m,T K
+m)m∈M
+a strong, consistent family for (F,E), F(Ω) a semi-Montel BC-space and U a set
+of uniqueness for (T K
+m,F)m∈M. If
+(i) E is complete, or
+(ii) E is sequentially complete and for every f ∈ FE′(U,E) and f ′ ∈ F(Ω)′
+there is a sequence (f ′
+n)n∈N in F(Ω)′ converging to f ′ in F(Ω)′
+κ such that
+Rt
+f(f ′
+n) ∈ J (E) for every n ∈ N,
+then the restriction map RU,E′∶S(F(Ω)εE) → FE′(U,E) is surjective.
+Proof. Let f ∈ FE′(U,E).
+As in Theorem 3.2.4 we equip J (E) with the
+system of seminorms given by
+pB○α(J (x)) ∶= sup
+e′∈B○α
+∣J (x)(e′)∣ = pα(x),
+x ∈ E,
+(45)
+for all α ∈ A where Bα ∶= {x ∈ E ∣ pα(x) < 1}. We claim Rt
+f ∈ L(F(Ω)′
+κ,J (E)).
+Indeed, we have for y ∈ F(Ω)′
+pB○α(Rt
+f(y)) = sup
+e′∈B○α
+∣y(fe′)∣ =
+sup
+x∈Rf (B○α)
+∣y(x)∣ ≤ sup
+x∈Kα
+∣y(x)∣
+(46)
+where Kα ∶= Rf(B○α).
+Due to Proposition 5.2.14 the set Rf(B○
+α) is absolutely
+convex and relatively compact, implying that Kα is absolutely convex and compact
+in F(Ω) by [89, 6.2.1 Proposition, p. 103]. Further, we have for all e′ ∈ E′ and
+(m,x) ∈ U
+Rt
+f(T K
+m,x)(e′) = T K
+m,x(fe′) = (e′ ○ f)(m,x) = J (f(m,x))(e′)
+(47)
+and thus Rt
+f(T K
+m,x) ∈ J (E).
+First, let condition (i) be satisfied, i.e. let E be complete, and f ′ ∈ F(Ω)′. The
+span of {T K
+m,x ∣ (m,x) ∈ U} is dense in F(Ω)′
+κ since U is a set of uniqueness for
+
+80
+5. APPLICATIONS
+F(Ω). Thus there is a net (f ′
+τ) converging to f ′ in FV(Ω)′
+κ with Rt
+f(f ′
+τ) ∈ J (E)
+and
+pB○α(Rt
+f(f ′
+τ) − Rt
+f(f ′)) ≤
+(46) sup
+x∈Kα
+∣(f ′
+τ − f ′)(x)∣ → 0
+(48)
+for all α ∈ A. We gain that (Rt
+f(f ′
+τ)) is a Cauchy net in the complete space J (E).
+Hence it has a limit g ∈ J (E) which coincides with Rt
+f(f ′) since
+pB○α(g − Rt
+f(f ′)) ≤ pB○α(g − Rt
+f(f ′
+τ)) + pB○α(Rt
+f(f ′
+τ) − Rt
+f(f ′))
+≤
+(48)pB○α(g − Rt
+f(f ′
+τ)) + sup
+x∈Kα
+∣(f ′
+τ − f ′)(x)∣ → 0
+for all α ∈ A. We conclude that Rt
+f(f ′) ∈ J (E) for every f ′ ∈ F(Ω)′.
+Second, let condition (ii) be satisfied and f ′ ∈ F(Ω)′. Then there is a sequence
+(f ′
+n) in F(Ω)′ converging to f ′ in F(Ω)′
+κ such that Rt
+f(f ′
+n) ∈ J (E) for every
+n ∈ N. From (46) we derive that (Rt
+f(f ′
+n)) is a Cauchy sequence in the sequentially
+complete space J (E) converging to Rt
+f(f ′) ∈ J (E).
+Therefore we obtain in both cases that Rt
+f ∈ L(F(Ω)′
+κ,J (E)). So we get for
+all α ∈ A and y ∈ F(Ω)′
+pα((J −1 ○ Rt
+f)(y)) =
+(45) pB○α(J ((J −1 ○ Rt
+f)(y))) = pB○α(Rt
+f(y)) ≤
+(46) sup
+x∈Kα
+∣y(x)∣.
+This implies J −1 ○ Rt
+f ∈ L(F(Ω)′
+κ,E) = F(Ω)εE (as linear spaces). We set F ∶=
+S(J −1 ○ Rt
+f) and obtain from consistency that
+T E
+m(F)(x) = T E
+mS(J −1 ○Rt
+f)(x) = J −1(Rt
+f(T K
+m,x)) =
+(47) J −1(J (f(m,x))) = f(m,x)
+for every (m,x) ∈ U, which means RU,E′(F) = f.
+□
+If E is complete and U a set of uniqueness for (T K
+m,F)m∈M with {T K
+m,x ∣ (m,x) ∈
+U} = {δx ∣ x ∈ Λ}, Λ ⊂ Ω, then we get [77, 0.1, p. 217] as a special case. Condition
+(i) and (ii) are adaptations of Condition 3.2.3 a) and c) from FV(Ω,E) and Rf to
+FE′(U,E) and Rf. We also treat an adaptation of Condition 3.2.3 e) in Theorem
+5.2.52. Condition 3.2.3 b) and d) may be adapted as well but we restrict to the
+ones we actually apply. First, we apply Theorem 5.2.15 to the space of bounded
+zero-solutions of a hypoelliptic linear partial differential operator equipped with the
+strict topology β from Proposition 4.2.24.
+5.2.16. Proposition. Let Ω ⊂ Rd be open and P(∂)K a hypoelliptic linear
+partial differential operator.
+Then (C∞
+P (∂),b(Ω),β) is a B-complete semi-Montel
+space.
+Proof. Due to the proof of Proposition 4.2.24 we know that β coincides with
+the mixed topology γ(τc,∥ ⋅ ∥∞). It is easy to check that the closed ∥ ⋅ ∥∞-unit ball
+B∥⋅∥∞ is τc-compact in C∞
+P (∂),b(Ω). Thus [46, Section I.1, 1.13 Proposition, p. 11]
+yields that (C∞
+P (∂),b(Ω),β) is a semi-Montel space. From [150, 2.9 Theorem, p. 185]
+it follows that the space is B-complete.
+□
+5.2.17. Corollary. Let Ω ⊂ Rd be open, E a complete lcHs, P(∂)K a hypoel-
+liptic linear partial differential operator, (T E
+m,T K
+m)m∈M a strong, consistent family
+for ((C∞
+P (∂),b(Ω),β),E) and U a set of uniqueness for (T K
+m,(C∞
+P (∂),b(Ω),β))m∈M.
+If f∶U → E is a function such that there is fe′ ∈ C∞
+P (∂),b(Ω) for each e′ ∈ E′
+with T K
+m(fe′)(x) = (e′ ○ f)(m,x) for all (m,x) ∈ U, then there is a unique F ∈
+C∞
+P (∂),b(Ω,E) with T E
+m(F)(x) = f(m,x) for all (m,x) ∈ U.
+
+5.2. EXTENSION OF VECTOR-VALUED FUNCTIONS
+81
+Proof. The space (C∞
+P (∂),b(Ω),β) is a semi-Montel BC-space by Proposition
+5.2.16. Moreover, (C∞
+P (∂),b(Ω),β) and (C∞
+P (∂),b(Ω,E),β) are ε-compatible by Propo-
+sition 4.2.24, yielding our statement by Theorem 5.2.15 (i) and Proposition 5.2.8.
+□
+Especially, for any m ∈ N0 the family ((∂β)E,(∂β)K)β∈Nd
+0,∣β∣≤m is strong and con-
+sistent for ((C∞
+P (∂),b(Ω),β),E) by the proof of Proposition 4.2.24. It is always pos-
+sible to construct a strong, consistent family (T E
+m,T K
+m)m∈M for ((C∞
+P (∂),b(Ω),β),E)
+from a given set of uniqueness (T K
+m,(C∞
+P (∂),b(Ω),β))m∈M due to Remark 4.3.11 b)
+and c).
+Similarly, we may apply Theorem 5.2.15 to the space E{Mp}(Ω,E) of ultradiffer-
+entiable functions of class {Mp} of Roumieu-type from Example 3.1.9 f). E{Mp}(Ω)
+is a projective limit of a countable sequence of DFS-spaces by [99, Theorem 2.6,
+p. 44] and thus webbed because being webbed is stable under the formation of
+projective and inductive limits of countable sequences by [89, 5.3.3 Corollary, p.
+92]. Further, if the sequence (Mp)p∈N0 satisfies Komatsu’s conditions (M.1) and
+(M.3)’, then E{Mp}(Ω) is a Montel space by [99, Theorem 5.12, p. 65–66]. The
+spaces E{Mp}(Ω) and E{Mp}(Ω,E) are ε-compatible if (M.1) and (M.3)’ hold and
+E is complete by Example 3.2.11 b). Hence Theorem 5.2.15 (i) is applicable.
+5.2.18. Remark. We remark that Remark 5.2.6 and Theorem 5.2.15 still hold
+if the map S∶F(Ω)εE → F(Ω,E) is only a linear isomorphism into, i.e. an isomor-
+phism into of linear spaces, since the topological nature of ε-into-compatibility is
+not used in the proof. In particular, this means that it can be applied to the space
+M(Ω,E) of meromorphic functions on an open, connected set Ω ⊂ C with values
+in an lcHs E over C (see [29, p. 356]). The space M(Ω) is a Montel LF-space,
+thus webbed by [89, 5.3.3 Corollary (b), p. 92], due to the proof of [80, Theorem
+3 (a), p. 294–295] if it is equipped with the locally convex topology τML given in
+[80, p. 292]. By [29, Proposition 6, p. 357] the map S∶M(Ω)εE → M(Ω,E) is
+a linear isomorphism if E is locally complete and does not contain the space CN.
+Therefore we can apply Theorem 5.2.15 if E is complete and does not contain CN.
+This augments [92, Theorem 12, p. 12] where E is assumed to be locally complete
+with suprabarrelled strong dual and (T E,T C) = (idEΩ,idCΩ).
+F(Ω) a Fréchet–Schwartz space and E locally complete. We recall the
+following abstract extension result.
+5.2.19. Proposition ([30, Proposition 7, p. 231]). Let E be a locally complete
+lcHs, Y a Fréchet–Schwartz space, X ⊂ Y ′
+b (= Y ′
+κ) dense and A∶X → E linear. Then
+the following assertions are equivalent:
+a) There is a (unique) extension ̂A ∈ Y εE of A.
+b) (At)−1(Y ) (= {e′ ∈ E′ ∣ e′ ○ A ∈ Y }) determines boundedness in E.
+Next, we generalise [30, Theorem 9, p. 232] using the preceding proposition.
+The proof of the generalisation is simply obtained by replacing the set of uniqueness
+in the proof of [30, Theorem 9, p. 232] by our more general set of uniqueness.
+5.2.20. Theorem. Let E be a locally complete lcHs, G ⊂ E′ determine bound-
+edness and F(Ω) and F(Ω,E) be ε-into-compatible. Let (T E
+m,T K
+m)m∈M be a strong,
+consistent family for (F,E), F(Ω) a Fréchet–Schwartz space and U a set of unique-
+ness for (T K
+m,F)m∈M. Then the restriction map RU,G∶S(F(Ω)εE) → FG(U,E) is
+surjective.
+
+82
+5. APPLICATIONS
+Proof. Let f ∈ FG(U,E).
+We choose X ∶= span{T K
+m,x ∣ (m,x) ∈ U} and
+Y ∶= F(Ω). Let A∶X → E be the linear map generated by A(T K
+m,x) ∶= f(m,x).
+The map A is well-defined since G is σ(E′,E)-dense. Let e′ ∈ G and fe′ be the
+unique element in F(Ω) such that T K
+m(fe′)(x) = (e′ ○ A)(T K
+m,x) for all (m,x) ∈
+U.
+This equation allows us to consider fe′ as a linear form on X (by setting
+fe′(T K
+m,x) ∶= (e′ ○ A)(T K
+m,x)), which yields e′ ○ A ∈ F(Ω) for all e′ ∈ G. It follows
+that G ⊂ (At)−1(Y ), implying that (At)−1(Y ) determines boundedness. Applying
+Proposition 5.2.19, there is an extension ̂A ∈ F(Ω)εE of A and we set F ∶= S(̂A).
+We note that
+T E
+m(F)(x) = T E
+mS(̂A)(x) = ̂A(T K
+m,x) = A(T K
+m,x) = f(m,x)
+for all (m,x) ∈ U by consistency, yielding RU,G(F) = f.
+□
+Let us apply the preceding theorem to our weighted spaces of continuously
+partially differentiable functions and its subspaces from Example 3.1.9 and Example
+4.2.22.
+5.2.21. Corollary. Let E be a locally complete lcHs, G ⊂ E′ determine bound-
+edness, V∞ a directed family of weights which is locally bounded away from zero on
+an open set Ω ⊂ Rd, let F(Ω) be a Fréchet–Schwartz space and U ⊂ Nd
+0 × Ω a set
+of uniqueness for (∂β,F)β∈Nd
+0 where F stands for CV∞, CV∞
+0 , CV∞
+P (∂) or CV∞
+P (∂),0.
+Then the following holds:
+a) If f∶U → E is a function such that there is fe′ ∈ F(Ω) for each e′ ∈ G
+with ∂βfe′(x) = (e′ ○ f)(β,x) for all (β,x) ∈ U, then there is a unique
+F ∈ F(Ω,E) with (∂β)EF(x) = f(β,x) for all (β,x) ∈ U.
+b) If U ⊂ Ω and f∶U → E is a function such that e′ ○ f admits an extension
+fe′ ∈ F(Ω) for every e′ ∈ G, then there is a unique extension F ∈ F(Ω,E)
+of f.
+c) F(Ω,E) = {f∶Ω → E ∣ ∀ e′ ∈ G ∶ e′ ○ f ∈ F(Ω)}.
+Proof. The strength and consistency of ((∂β)E,∂β)β∈Nd
+0 for (F,E) and the ε-
+compatibility of F(Ω) and F(Ω,E) follow from Example 3.2.7 e)+f) and Example
+4.2.22 c)+f). This implies that part a) and its special case part b) hold by Theorem
+5.2.20 and Proposition 5.2.8. Part c) follows from part b) and Proposition 5.2.10
+since U ∶= Ω is a set of uniqueness for (idKΩ,F).
+□
+5.2.22. Remark. Let V∞ be a directed family of weights which is locally
+bounded away from zero on an open set Ω ⊂ Rd.
+a) Then any dense set U ⊂ Ω is a set of uniqueness for (idKΩ,F) with F =
+CV∞, CV∞
+0 , CV∞
+P (∂) or CV∞
+P (∂),0 due to continuity.
+b) Let Ω be connected and x0 ∈ Ω.
+Then U ∶= {(en,x) ∣ 1 ≤ n ≤ d, x ∈
+Ω} ∪ {(0,x0)} is a set of uniqueness for (∂β,F)β∈N0 by the mean value
+theorem with F from a).
+c) Let K ∶= R, d ∶= 1, Ω ∶= (a,b) ⊂ R, g∶(a,b) → N and x0 ∈ (a,b). Then
+U ∶= {(g(x),x) ∣ x ∈ (a,b)} ∪ {(n,x0) ∣ n ∈ N0} is a set of uniqueness for
+(∂β,F)β∈N0 with F from a). Indeed, if f ∈ F(Ω) and 0 = ∂g(x)f(x) for all
+x ∈ (a,b), then f is a polynomial by [56, Chap. 11, Theorem, p. 53]. If, in
+addition, 0 = ∂nf(x0) for all n ∈ N0, then the polynomial f must vanish
+on the whole interval Ω.
+d) Let Ω ⊂ C be connected. Then any set U ⊂ Ω with an accumulation point
+in Ω is a set of uniqueness for (idCΩ,CV∞
+∂ ) by the identity theorem for
+holomorphic functions.
+
+5.2. EXTENSION OF VECTOR-VALUED FUNCTIONS
+83
+e) Let Ω ⊂ C be connected and z0 ∈ Ω. Then U ∶= {(n,z0) ∣ n ∈ N0} is a set
+of uniqueness for (∂n
+C,CV∞
+∂ )n∈N0 by local power series expansion and the
+identity theorem.
+f) Let Ω ⊂ Rd be connected.
+Then any non-empty open set U ⊂ Ω is a
+set of uniqueness for (idKΩ,CV∞
+∆) by the identity theorem for harmonic
+functions (see e.g. [85, Theorem 5, p. 218]).
+g) Further examples of sets of uniqueness for (idKΩ,CV∞
+∆) are given in [98].
+In part e) a special case of Remark 5.2.3 b) is used, namely, that CW∞
+∂ (Dr(z0))
+has a Schauder basis with associated coefficient functionals (δz0 ○ ∂n
+C)n∈N0 where
+0 < r ≤ ∞ is such that Dr(z0) ⊂ Ω. In order to obtain some sets of uniqueness which
+are more sensible w.r.t. the family of weights V∞, we turn to entire and harmonic
+functions fulfilling some growth conditions. For a family V ∶= (νj)j∈N of continuous
+weights on Rd set V∞ ∶= (νj,m)j∈N,m∈N0 where νj,m∶{β ∈ Nd
+0 ∣ ∣β∣ ≤ m}×Rd → [0,∞),
+νj,m(β,x) ∶= νj(x). We know that CVP (∂)(Rd,E) = CV∞
+P (∂)(Rd,E) as locally convex
+spaces and that CVP (∂)(Rd) is a nuclear Fréchet space for P(∂) = ∂ or P(∂) = ∆
+by Proposition 4.2.19 if E is a locally complete lcHs and V fulfils Condition 4.2.18.
+In particular, Condition 4.2.18 is fulfilled if νj(x) ∶= exp(−(τ + 1
+j )∣x∣), x ∈ Rd, for
+all j ∈ N and some 0 ≤ τ < ∞ and thus we can apply Corollary 5.2.21 to the
+spaces Aτ
+∂(C,E) = CV∂(C,E) of entire and Aτ
+∆(Rd,E) = CV∆(Rd,E) of harmonic
+functions of exponential type τ by Remark 4.2.20. Hence we may complement our
+list in Remark 5.2.22 by some more examples for spaces of functions of exponential
+type 0 ≤ τ < ∞.
+5.2.23. Remark. The following sets U ⊂ C are sets of uniqueness for (idCC,Aτ
+∂).
+a) If τ < π, then U ∶= N0 is a set of uniqueness by [21, 9.2.1 Carlson’s theorem,
+p. 153].
+b) Let δ > 0 and (λn)n∈N ⊂ (0,∞) such that λn+1 − λn > δ for all n ∈ N. Then
+U ∶= (λn)n∈N is a set of uniqueness if limsupr→∞ r−2τ/πψ(r) = ∞ where
+ψ(r) ∶= exp(∑λn<r λ−1
+n ), r > 0, by [21, 9.5.1 Fuchs’s theorem, p. 157–158].
+The following sets U are sets of uniqueness for (∂n
+C,Aτ
+∂)n∈N0.
+c) Let (λn)n∈N0 ⊂ C with ∣λn∣ < 1 for all n ∈ N0. If τ < ln(2), then U ∶=
+{(n,λn) ∣ n ∈ N0} is a set of uniqueness by [21, 9.11.1 Theorem, p. 172].
+If τ < ln(2 +
+√
+3), then U ∶= {(2n + 1,0) ∣ n ∈ N0} ∪ {(2n,λn) ∣ n ∈ N0} is a
+set of uniqueness by [21, 9.11.3 Theorem, p. 173].
+d) Let (λn)n∈N0 ⊂ C with limsupn→∞ n−1 ∑n
+k=1 ∣λk∣ ≤ 1.
+If τ < e−1, then
+U ∶= {(n,λn) ∣ n ∈ N0} is a set of uniqueness by [21, 9.11.4 Theorem, p.
+173].
+The following sets U ⊂ Rd are sets of uniqueness for (idRRd ,Aτ
+∆).
+e) Let d ∶= 2. If there is k ∈ N with τ < π/k, then U ∶= Z ∪ (Z + ik) is a set of
+uniqueness by [22, Theorem 1, p. 425].
+f) Let d ∶= 2. If τ < π and θ ∉ πQ, then U ∶= Z ∪ (eiθZ) is a set of uniqueness
+by [22, Theorem 2, p. 426].
+g) If τ < π, then U ∶= {0,1} × Zd−1 is a set of uniqueness by [145, Corollary
+1.8, p. 312].
+h) If τ < π and a ∈ R with ∣a∣ ≤
+√
+1/(d − 1), then U ∶= Zd−1 × {0,a} is a set of
+uniqueness by [185, Theorem A, p. 335].
+i) Further examples of sets of uniqueness can be found in [10].
+The following sets U are sets of uniqueness for ((∂β)R,Aτ
+∆)β∈Nd
+0.
+j) If τ < π, then U ∶= {(β,(x,0)) ∣ β ∈ {0,ed}, x ∈ Zd−1} is a set of uniqueness
+by [185, Theorem B, p. 335]. Further examples can be found in [10].
+
+84
+5. APPLICATIONS
+We need the following weak-strong principle in our last section for the space
+E0(E) of E-valued infinitely continuously partially differentiable functions on (0,1)
+such that all derivatives can be continuously extended to the boundary and vanish
+at 1.
+5.2.24. Corollary. Let E be a locally complete lcHs and G ⊂ E′ determine
+boundedness. Then E0(E) = {f∶(0,1) → E ∣ ∀ e′ ∈ G ∶ e′ ○ f ∈ E0}.
+Proof. By Example 4.2.29 E0 is a Fréchet–Schwartz space and E0 and E0(E)
+are ε-compatible. We derive our statement from Theorem 5.2.20 and Proposition
+5.2.10 with (T E,T K) ∶= (idEU ,idKU ) and U ∶= (0,1).
+□
+Fν(Ω) a Banach space and E locally complete. In this subsection we
+consider function spaces F(Ω,E) with a certain structure, namely, spaces FV(Ω,E)
+from Definition 3.1.4 where the family of weights V = (νj,m)j∈J,m∈M only consists
+of one weight function, i.e. the sets J and M can be chosen as singletons. So for
+two non-empty sets Ω and ω, a weight ν∶ω → (0,∞), a linear operator T E∶EΩ ⊃
+domT E → Eω and a linear subspace AP(Ω,E) of EΩ we consider the space
+Fν(Ω,E) = {f ∈ F(Ω,E) ∣ ∀ α ∈ A ∶ ∣f∣α < ∞}
+where
+F(Ω,E) = AP(Ω,E) ∩ domT E
+and
+∣f∣α = ∣f∣Fν(Ω),α = sup
+x∈ω pα(T E(f)(x))ν(x).
+For instance, if Ω ∶= ω, T E ∶= idEΩ and ν ∶= 1 on Ω, then Fν(Ω,E) is the
+linear subspace of F(Ω,E) consisting of bounded functions. We use the methods
+developed in [70, 93] where, in particular, the special case that Fν(Ω) is the space
+of bounded smooth functions on an open set Ω ⊂ Rd in the kernel of a hypoelliptic
+linear partial differential operator resp. a weighted space of holomorphic functions
+on an open subset Ω of a Banach space is treated. The lack of compact subsets of an
+infinite dimensional Banach space Fν(Ω) is compensated in [70, 93] by equipping
+F(Ω) with a locally convex Hausdorff topology such that the closed unit ball of
+Fν(Ω) is compact in F(Ω). Among others, the space F(Ω,E) ∶= (O(Ω,E),τc) of
+holomorphic functions on an open set Ω ⊂ C with values in a locally complete space
+E equipped with topology τc of compact convergence is used in [70] and the space
+Fν(Ω,E) ∶= H∞(Ω,E) of E-valued bounded holomorphic functions on Ω.
+5.2.25. Proposition. Let F(Ω) and F(Ω,E) be ε-into-compatible, (T E,T K)
+a consistent family for (F,E) and a generator for (Fν,E) and the map i∶Fν(Ω) →
+F(Ω), f ↦ f, continuous. We set
+Fεν(Ω,E) ∶= S({u ∈ F(Ω)εE ∣ u(B○F (Ω)′
+Fν(Ω) ) is bounded in E})
+where B○F (Ω)′
+Fν(Ω) ∶= {y′ ∈ F(Ω)′ ∣ ∀ f ∈ BFν(Ω) ∶ ∣y′(f)∣ ≤ 1} and BFν(Ω) is the closed
+unit ball of Fν(Ω). Then the following holds:
+a) Fν(Ω) is a dom-space.
+b) Let u ∈ F(Ω)εE. Then
+sup
+y′∈B○F (Ω)′
+Fν(Ω)
+pα(u(y′)) = ∣S(u)∣Fν(Ω),α,
+α ∈ A.
+In particular,
+Fεν(Ω,E) = S({u ∈ F(Ω)εE ∣ ∀ α ∈ A ∶ ∣S(u)∣Fν(Ω),α < ∞}).
+
+5.2. EXTENSION OF VECTOR-VALUED FUNCTIONS
+85
+c) S(Fν(Ω)εE) ⊂ Fεν(Ω,E) ⊂ Fν(Ω,E) as linear spaces.
+If F(Ω) and
+F(Ω,E) are even ε-compatible, then Fεν(Ω,E) = Fν(Ω,E).
+d) If Fν(Ω,E) is Hausdorff, then
+(i) (T E,T K) is a consistent generator for (Fν,E).
+(ii) Fν(Ω) and Fν(Ω,E) are ε-into-compatible.
+(iii) (T E,T K) is a strong generator for (Fν,E) if it is a strong family for
+(F,E).
+Proof. Part a) follows from the continuity of the map i and the ε-into-
+compatibility of F(Ω) and F(Ω,E). Let us turn to part b). As in Lemma 3.1.8 it
+follows from the bipolar theorem that
+B○F (Ω)′
+Fν(Ω) = acx{T K
+x (⋅)ν(x) ∣ x ∈ ω},
+where acx denotes the closure w.r.t. κ(F(Ω)′,Fν(Ω)) of the absolutely convex hull
+acx of the set D ∶= {T K
+x (⋅)ν(x) ∣ x ∈ ω} on the right-hand side, and that
+sup
+y′∈B○F (Ω)′
+Fν(Ω)
+pα(u(y′)) =
+sup
+y′∈acx(D)
+pα(u(y′)) = sup
+y′∈D
+pα(u(y′)) = sup
+x∈ω pα(u(T K
+x ))ν(x)
+= sup
+x∈ω pα(T E(S(u))(x))ν(x) = ∣S(u)∣Fν(Ω),α
+by consistency, which proves part b).
+Let us address part c).
+The continuity of the map i implies the continuity
+of the inclusion Fν(Ω)εE ↪ F(Ω)εE and thus we obtain u∣F (Ω)′ ∈ F(Ω)εE for
+every u ∈ Fν(Ω)εE. If u ∈ Fν(Ω)εE and α ∈ A, then there are C0,C1 > 0 and an
+absolutely convex compact set K ⊂ Fν(Ω) such that K ⊂ C1BFν(Ω) and
+sup
+y′∈B○F (Ω)′
+Fν(Ω)
+pα(u(y′)) ≤ C0
+sup
+y′∈B○F (Ω)′
+Fν(Ω)
+sup
+f∈K
+∣y′(f)∣ ≤ C0C1,
+which implies S(Fν(Ω)εE) ⊂ Fεν(Ω,E). If f ∶= S(u) ∈ Fεν(Ω,E) and α ∈ A, then
+S(u) ∈ F(Ω,E) and
+∣f∣Fν(Ω),α = sup
+x∈ω pα(u(T K
+x )ν(x)) < ∞
+by consistency, yielding Fεν(Ω,E) ⊂ Fν(Ω,E). If F(Ω) and F(Ω,E) are even
+ε-compatible, then S(F(Ω)εE) = F(Ω,E), which yields Fεν(Ω,E) = Fν(Ω,E) by
+part b).
+Let us turn to part d). By part a) Fν(Ω) is a dom-space. Since Fν(Ω,E) is
+Hausdorff, it is also a dom-space due to Remark 3.1.6 c). We have u∣F (Ω)′ ∈ F(Ω)εE
+for every u ∈ Fν(Ω)εE and
+SFν(Ω)(u)(x) = u(δx) = u∣F (Ω)′(δx) = SF (Ω)(u∣F (Ω)′)(x),
+x ∈ Ω.
+In combination with S(F(Ω)εE) ⊂ F(Ω,E) and the consistency of (T E,T K) for
+(F,E) this yields that (T E,T K) is a consistent generator for (Fν,E). Thus part (i)
+holds and implies part (ii) by Theorem 3.1.12. If (T E,T K) is in addition a strong
+family for (F,E), then the inclusion Fν(Ω,E) ⊂ F(Ω,E) implies that e′ ○ f ∈
+F(Ω,E) and T K(e′ ○f)(x) = (e′ ○T E(f))(x) for all e′ ∈ E′, f ∈ Fν(Ω,E) and x ∈ ω.
+It follows that (T E,T K) is a strong generator for (Fν,E).
+□
+The canonical situation in part c) is that Fεν(Ω,E) and Fν(Ω,E) coincide
+as linear spaces for locally complete E as we will encounter in the forthcoming
+examples, e.g. if Fν(Ω,E) ∶= H∞(Ω,E) and F(Ω,E) ∶= (O(Ω,E),τc) for an open
+set Ω ⊂ C. That all three spaces in part c) coincide is usually only guaranteed
+by Corollary 3.2.5 (iii) if E is a semi-Montel space. Therefore the ‘mingle-mangle’
+space Fεν(Ω,E) is a good replacement for S(Fν(Ω)εE) for our purpose.
+
+86
+5. APPLICATIONS
+5.2.26. Remark. Let (T E,T K) be a strong, consistent family for (F,E) and
+a generator for (Fν,E).
+Let F(Ω) and F(Ω,E) be ε-into-compatible and the
+inclusion Fν(Ω) ↪ F(Ω) continuous. Consider a set of uniqueness U for (T K,Fν)
+and a separating subspace G ⊂ E′. For u ∈ F(Ω)εE such that u(B○F (Ω)′
+Fν(Ω) ) is bounded
+in E, i.e. S(u) ∈ Fεν(Ω,E), we set f ∶= S(u). Then f ∈ F(Ω,E) by the ε-into-
+compatibility and we define ̃f∶U → E, ̃f(x) ∶= T E(f)(x). This yields
+(e′ ○ ̃f)(x) = (e′ ○ T E(f))(x) = T K(e′ ○ f)(x)
+(49)
+for all x ∈ U and fe′ ∶= e′ ○ f ∈ F(Ω) for each e′ ∈ E′ by the strength of the family.
+Moreover, T K
+x (⋅)ν(x) ∈ B○F (Ω)′
+Fν(Ω) for every x ∈ ω, which implies that for every e′ ∈ E′
+there are α ∈ A and C > 0 such that
+∣fe′∣Fν(Ω) = sup
+x∈ω
+∣e′(u(T K
+x (⋅)ν(x))∣ ≤ C
+sup
+y′∈B○F (Ω)′
+Fν(Ω)
+pα(u(y′)) < ∞
+by strength and consistency. Hence fe′ ∈ Fν(Ω) for every e′ ∈ E′ and ̃f ∈ FνG(U,E).
+Under the assumptions of Remark 5.2.26 the map
+RU,G∶Fεν(Ω,E) → FνG(U,E), f ↦ (T E(f)(x))x∈U,
+(50)
+is well-defined and linear. In addition, we derive from (49) that RU,G is injective
+since U is a set of uniqueness and G ⊂ E′ separating. The replacement of Question
+5.2.9 reads as follows.
+5.2.27. Question. Let the assumptions of Remark 5.2.26 be fulfilled. When is
+the injective restriction map
+RU,G∶Fεν(Ω,E) → FνG(U,E), f ↦ (T E(f)(x))x∈U,
+surjective?
+Due to Proposition 5.2.25 c) the Question 5.2.1 is a special case of this question
+if Λ ⊂ Ω =∶ ω and U ∶= Λ is a set of uniqueness for (idKΩ,Fν). We recall the following
+extension result for continuous linear operators.
+5.2.28. Proposition ([70, Proposition 2.1, p. 691]). Let E be a locally complete
+lcHs, G ⊂ E′ determine boundedness, Z a Banach space whose closed unit ball BZ
+is a compact subset of an lcHs Y and X ⊂ Y ′ be a σ(Y ′,Z)-dense subspace. If
+A∶X → E is a σ(X,Z)-σ(E,G)-continuous linear map, then there exists a (unique)
+extension ̂A ∈ Y εE of A such that ̂A(B○Y ′
+Z
+) is bounded in E where B○Y ′
+Z
+∶= {y′ ∈
+Y ′ ∣ ∀ z ∈ BZ ∶ ∣y′(z)∣ ≤ 1}.
+Now, we are able to generalise [70, Theorem 2.2, p. 691] and [93, Theorem 10,
+p. 5].
+5.2.29. Theorem. Let E be a locally complete lcHs, G ⊂ E′ determine bounded-
+ness and F(Ω) and F(Ω,E) be ε-into-compatible. Let (T E,T K) be a generator for
+(Fν,E) and a strong, consistent family for (F,E), Fν(Ω) a Banach space whose
+closed unit ball BFν(Ω) is a compact subset of F(Ω) and U a set of uniqueness for
+(T K,Fν). Then the restriction map
+RU,G∶Fεν(Ω,E) → FνG(U,E)
+is surjective.
+Proof. Let f ∈ FνG(U,E). We set X ∶= span{T K
+x ∣ x ∈ U}, Y ∶= F(Ω) and
+Z ∶= Fν(Ω). The consistency of (T E,T K) for (F,E) yields that X ⊂ Y ′. From
+U being a set of uniqueness of Z follows that X is σ(Z′,Z)-dense. Since BZ is a
+compact subset of Y , it follows that Z is a linear subspace of Y and the inclusion
+
+5.2. EXTENSION OF VECTOR-VALUED FUNCTIONS
+87
+Z ↪ Y is continuous, which yields y′
+∣Z ∈ Z′ for every y′ ∈ Y ′. Thus X is σ(Y ′,Z)-
+dense. Let A∶X → E be the linear map determined by A(T K
+x ) ∶= f(x). The map A
+is well-defined since G is σ(E′,E)-dense. Due to
+e′(A(T K
+x )) = (e′ ○ f)(x) = T K
+x (fe′)
+for every e′ ∈ G and x ∈ U we have that A is σ(X,Z)-σ(E,G)-continuous. We
+apply Proposition 5.2.28 and gain an extension ̂A ∈ Y εE of A such that ̂A(B○Y ′
+Z
+) is
+bounded in E. We set ̃F ∶= S(̂A) ∈ Fεν(Ω,E) and get for all x ∈ U that
+T E( ̃F)(x) = T ES(̂A)(x) = ̂A(T K
+x ) = f(x)
+by consistency for (F,E), implying RU,G( ̃F) = f.
+□
+Let Ω ⊂ Rd be open, E an lcHs and P(∂)E∶C∞(Ω,E) → C∞(Ω,E) a linear
+partial differential operator which is hypoelliptic if E = K. We consider the weighted
+space CVP (∂)(Ω,E) of zero solutions from Proposition 4.2.14 where the familiy of
+weights V only consists of one continuous weight ν∶Ω → (0,∞), i.e. the space
+CνP (∂)(Ω,E) = {f ∈ C∞
+P (∂)(Ω,E) ∣ ∀ α ∈ A ∶ ∣f∣ν,α ∶= sup
+x∈Ω
+pα(f(x))ν(x) < ∞}.
+5.2.30. Corollary. Let E be a locally complete lcHs, G ⊂ E′ determine bound-
+edness, Ω ⊂ Rd open, P(∂)K a hypoelliptic linear partial differential operator,
+ν∶Ω → (0,∞) continuous and U a set of uniqueness for (idKΩ,CνP (∂)). If f∶U → E
+is a function such that e′ ○ f admits an extension fe′ ∈ CνP (∂)(Ω) for every e′ ∈ G,
+then there exists a unique extension F ∈ CνP (∂)(Ω,E) of f.
+Proof. We choose F(Ω) ∶= (C∞
+P (∂)(Ω),τc) and F(Ω,E) ∶= (C∞
+P (∂)(Ω,E),τc).
+Then we have Fν(Ω) = CνP (∂)(Ω) and Fν(Ω,E) = CνP (∂)(Ω,E) with the generator
+(T E,T K) ∶= (idEΩ,idKΩ) for (Fν,E).
+We note that F(Ω) and F(Ω,E) are ε-
+compatible and (T E,T K) is a strong, consistent family for (F,E) by Proposition
+4.2.17. We observe that Fν(Ω) is a Banach space by Proposition 4.2.14 and for
+every compact K ⊂ Ω we have
+sup
+x∈K
+∣f(x)∣ ≤
+(22) sup
+z∈K
+ν(z)−1∣f∣ν ≤ sup
+z∈K
+ν(z)−1,
+f ∈ BFν(Ω),
+yielding that BFν(Ω) is bounded in F(Ω). The space F(Ω) = (C∞
+P (∂)(Ω),τc) is a
+Fréchet–Schwartz space, thus a Montel space, and it is easy to check that BFν(Ω) is
+τc-closed. Hence the bounded and τc-closed set BFν(Ω) is compact in F(Ω). Finally,
+we remark that the ε-compatibility of F(Ω) and F(Ω,E) in combination with the
+consistency of (idEΩ,idKΩ) for (F,E) gives Fεν(Ω,E) = Fν(Ω,E) as linear spaces
+by Proposition 5.2.25 c). From Theorem 5.2.29 follows our statement.
+□
+If Ω = D ⊂ C is the open unit disc, P(∂) = ∂ the Cauchy–Riemann operator
+and ν = 1 on D, then CνP (∂)(Ω,E) = H∞(D,E) and a sequence U ∶= (zn)n∈N ⊂ D of
+distinct elements is a set of uniqueness for (idCD,H∞) if and only if it satisfies the
+Blaschke condition ∑n∈N(1 − ∣zn∣) = ∞ (see e.g. [149, 15.23 Theorem, p. 303]).
+For a continuous function ν∶D → (0,∞) and a complex lcHs E we define the
+Bloch type spaces
+Bν(D,E) ∶= {f ∈ O(D,E) ∣ ∀ α ∈ A ∶ ∣f∣ν,α < ∞}
+with
+∣f∣ν,α ∶= max(pα(f(0)),sup
+z∈D
+pα((∂1
+C)Ef(z))ν(z)).
+If E = C, we write f ′(z) ∶= (∂1
+C)Cf(z) for z ∈ D and f ∈ O(D).
+5.2.31. Proposition. If ν∶D → (0,∞) is continuous, then Bν(D) is a Banach
+space.
+
+88
+5. APPLICATIONS
+Proof. Let f ∈ Bν(D). From the estimates
+∣f(z)∣ ≤ ∣f(0)∣ + ∣
+z
+∫
+0
+f ′(ζ)dζ∣ ≤ ∣f(0)∣ +
+∣z∣
+minξ∈[0,z] ν(ξ) sup
+ζ∈[0,z]
+∣f ′(ζ)∣ν(ζ)
+≤ 2max(1,
+∣z∣
+minξ∈[0,z] ν(ξ))∣f∣ν
+for every z ∈ D and
+max
+∣z∣≤r ∣f(z)∣ ≤ 2max(1,
+r
+min∣z∣≤r ν(z))∣f∣ν
+(51)
+for all 0 < r < 1 and f ∈ Bν(D) it follows that Bν(D) is a Banach space by using
+the completeness of (O(D),τc) analogously to the proof of Proposition 4.2.14.
+□
+5.2.32. Proposition. Let Ω ⊂ C be open and E a locally complete lcHs over
+C. Then ((∂n
+C)E,(∂n
+C)C)n∈N0 is a strong, consistent family for ((O(Ω),τc),E).
+Proof. We recall from (5) that the real and complex derivatives are related
+by
+(∂β)Ef(z) = iβ2(∂∣β∣
+C )Ef(z),
+z ∈ Ω,
+(52)
+for every f ∈ O(Ω,E) and β = (β1,β2) ��� N2
+0. Further, the Fréchet space (O(Ω),τc)
+is barrelled. Due to Proposition 3.1.11 c) and (52) we have for all u ∈ (O(Ω),τc)εE
+(∂n
+C)ES(u)(z) = u(δz ○ (∂n
+C)C),
+n ∈ N0, z ∈ Ω,
+which means that ((∂n
+C)E,(∂n
+C)C)n∈N0 is consistent.
+Moreover, we have
+(∂n
+C)C(e′ ○ f)(z) = e′((∂n
+C)Ef(z)),
+n ∈ N0, z ∈ Ω,
+for all e′ ∈ E′ and f ∈ O(Ω,E), implying the strength of ((∂n
+C)E,(∂n
+C)C)n∈N0.
+□
+Let E be an lcHs and ν∶D → (0,∞) be continuous. We set ω ∶= {0}∪{(1,z) ∣ z ∈
+D}, define the operator T E∶O(D,E) → Eω by
+T E(f)(0) ∶= f(0)
+and
+T E(f)(1,z) ∶= (∂1
+C)Ef(z), z ∈ D,
+and the weight ν∗∶ω → (0,∞) by
+ν∗(0) ∶= 1
+and
+ν∗(1,z) ∶= ν(z), z ∈ D.
+Then we have for every α ∈ A that
+∣f∣ν,α = sup
+x∈ω pα(T E(f)(x))ν∗(x),
+f ∈ Bν(D,E),
+and with F(D,E) ∶= O(D,E) we observe that Fν∗(D,E) = Bν(D,E) with generator
+(T E,T C).
+5.2.33. Corollary. Let E be a locally complete lcHs, G ⊂ E′ determine bound-
+edness, ν∶D → (0,∞) continuous and U∗ ⊂ D have an accumulation point in D. If
+f∶{0} ∪ ({1} × U∗) → E is a function such that there is fe′ ∈ Bν(D) for each e′ ∈ G
+with fe′(0) = e′(f(0)) and f ′
+e′(z) = e′(f(1,z)) for all z ∈ U∗, then there exists a
+unique F ∈ Bν(D,E) with F(0) = f(0) and (∂1
+C)EF(z) = f(1,z) for all z ∈ U∗.
+Proof. We take F(D) ∶= (O(D),τc) and F(D,E) ∶= (O(D,E),τc). Then we
+have Fν∗(D) = Bν(D) and Fν∗(Ω,E) = Bν(D,E) with the weight ν∗ and generator
+(T E,T C) for (Fν∗,E) described above.
+The spaces F(D) and F(D,E) are ε-
+compatible by Proposition 4.2.17 in combination with (23), and the generator is
+a strong, consistent family for (F,E) by Proposition 5.2.32. Due to Proposition
+5.2.31 Fν∗(D) = Bν(D) is a Banach space and we deduce from (51) that BFν∗(D) is
+compact in the Montel space (O(D),τc). We note that the ε-compatibility of F(Ω)
+
+5.2. EXTENSION OF VECTOR-VALUED FUNCTIONS
+89
+and F(Ω,E) in combination with the consistency of (T E,T C) for (F,E) gives
+Fεν∗(D,E) = Fν∗(D,E) as linear spaces by Proposition 5.2.25 c). In addition,
+U ∶= {0} ∪ {(1,z) ∣ z ∈ U∗} is a set of uniqueness for (T C,Fν∗) by the identity
+theorem, proving our statement by Theorem 5.2.29.
+□
+E a Fréchet space. In this section we restrict to the case that E is a Fréchet
+space and G ⊂ E′ is generated by a sequence that fixes the topology in E.
+5.2.34. Definition ([30, Definition 12, p. 8]). Let Y be a Fréchet space. An
+increasing sequence (Bn)n∈N of bounded subsets of Y ′
+b fixes the topology in Y if
+(B○
+n)n∈N is a fundamental system of zero neighbourhoods of Y .
+5.2.35. Remark. Let Y be a Banach space. If B ⊂ Y ′
+b is bounded, i.e. bounded
+w.r.t. the operator norm, such that B fixes the topology in Y , i.e. B○ is bounded
+in Y , then B is called an almost norming subset. Examples of almost norming
+subspaces are given in [7, Remark 1.2, p. 780–781]. For instance, the set of point
+evaluations B ∶= {δ1/n ∣ n ∈ N} is almost norming for the Y ∶= H∞(D) ∶= C∞
+∂,b(D).
+5.2.36. Definition (sb-restriction space). Let E be a Fréchet space, (Bn) fix
+the topology in E and G ∶= span(⋃n∈N Bn). Let FV(Ω) be a dom-space, U a set of
+uniqueness for (T K
+m,FV)m∈M and set
+FVG(U,E)sb ∶= {f ∈ FVG(U,E) ∣ ∀ n ∈ N ∶ {fe′ ∣ e′ ∈ Bn} is bounded in FV(Ω)}.
+Let E be a Fréchet space, (Bn) fix the topology in E, G ∶= span(⋃n∈N Bn),
+(T E
+m,T K
+m)m∈M be a strong, consistent generator for (FV,E) and U a set of unique-
+ness for (T K
+m,FV)m∈M. For u ∈ FV(Ω)εE we have RU,G(f) ∈ FVG(U,E) with
+f ∶= S(u) by Remark 5.2.6 and for j ∈ J and m ∈ M
+sup
+e′∈Bn
+∣fe′∣j,m = sup
+e′∈Bn
+sup
+x∈ωm
+∣e′(T E
+m(f)(x)νj,m(x))∣ = sup
+e′∈Bn
+sup
+y∈Nj,m(f)
+∣e′(y)∣
+with Nj,m(f) ∶= {T E
+m(f)(x)νj,m(x) ∣ x ∈ ωm}. This set is bounded in E since
+sup
+y∈Nj,m(f)
+pα(f) = ∣f∣j,m,α < ∞
+for all α ∈ A, implying supe′∈Bn ∣fe′∣j,m < ∞ and RU,G(f) ∈ FVG(U,E)sb. Hence the
+injective linear map
+RU,G∶S(FV(Ω)εE) → FVG(U,E)sb, f ↦ (T E
+m(f)(x))(m,x)∈U,
+is well-defined.
+5.2.37. Question. Let E be a Fréchet space, (Bn) fix the topology in E
+and G ∶= span(⋃n∈N Bn). Let (T E
+m,T K
+m)m∈M be a strong, consistent generator for
+(FV,E) and U a set of uniqueness for (T K
+m,FV)m∈M. When is the injective restric-
+tion map
+RU,G∶S(FV(Ω)εE) → FVG(U,E)sb, f ↦ (T E
+m(f)(x))(m,x)∈U,
+surjective?
+5.2.38. Remark. Let E be a Fréchet space with increasing system of seminorms
+(pαn)n∈N, Bn ∶= B○
+αn where Bαn ∶= {x ∈ E ∣ pαn(x) < 1}, (T E
+m,T K
+m)m∈M a strong,
+consistent generator for (FV,E) and U a set of uniqueness for (T K
+m,FV)m∈M. If
+FV(Ω) is a BC-space, then FVE′(U,E)sb = FVE′(U,E) by Proposition 5.2.14.
+Hence Theorem 5.2.15 (i) answers Question 5.2.37 in this case.
+Let us turn to the case where G need not coincide with E′.
+
+90
+5. APPLICATIONS
+FV(Ω) a Fréchet–Schwartz space and E a Fréchet space. We recall the
+following result.
+5.2.39. Proposition ([69, Lemma 9, p. 504]). Let E be a Fréchet space, (Bn)
+fix the topology in E, Y a Fréchet–Schwartz space and X ⊂ Y ′
+b (= Y ′
+κ) a dense
+subspace.
+Set G ∶= span(⋃n∈N Bn) and let A∶X → E be a linear map which is
+σ(X,Y )-σ(E,G)-continuous and satisfies that At(Bn) is bounded in Y for each
+n ∈ N. Then A has a (unique) extension ̂A ∈ Y εE.
+Next, we improve [69, Theorem 1 ii), p. 501].
+5.2.40. Theorem. Let E be a Fréchet space, (Bn) fix the topology in E and G ∶=
+span(⋃n∈N Bn), (T E
+m,T K
+m)m∈M a strong, consistent generator for (FV,E), FV(Ω)
+a Fréchet–Schwartz space and U a set of uniqueness for (T K
+m,FV)m∈M. Then the
+restriction map RU,G∶S(FV(Ω)εE) → FVG(U,E)sb is surjective.
+Proof. Let f ∈ FVG(U,E)sb.
+We set X ∶= span{T K
+m,x ∣ (m,x) ∈ U} and
+Y ∶= FV(Ω). Let A∶X → E be the linear map determined by A(T K
+m,x) ∶= f(m,x)
+which is well-defined since G is σ(E′,E)-dense. From
+e′(A(T K
+m,x)) = (e′ ○ f)(m,x) = T K
+m,x(fe′)
+for every e′ ∈ G and (m,x) ∈ U it follows that A is σ(X,Y )-σ(E,G)-continuous and
+sup
+e′∈Bn
+∣At(e′)∣j,k = sup
+e′∈Bn
+∣fe′∣j,k < ∞
+for all j ∈ J, k ∈ M and n ∈ N. Due to Proposition 5.2.39 there is an extension
+̂A ∈ FV(Ω)εE of A. We set F ∶= S(̂A) and get for all (m,x) ∈ U that
+T E
+m(F)(x) = T E
+mS(̂A)(x) = ̂A(T K
+m,x) = f(m,x)
+by consistency, which means RU,G(F) = f.
+□
+5.2.41. Corollary. Let E be a Fréchet space, (Bn) fix the topology in E and
+G ∶= span(⋃n∈N Bn).
+Let V ∶= (νj)j∈N be an increasing family of weights which
+is locally bounded away from zero on an open set Ω ⊂ Rd, P(∂)K a hypoelliptic
+linear partial differential operator, CVP (∂)(Ω) a Schwartz space and U ⊂ Ω a set of
+uniqueness for (idKΩ,CVP (∂)). If f∶U → E is a function such that e′ ○ f admits
+an extension fe′ ∈ CVP (∂)(Ω) for each e′ ∈ G and {fe′ ∣ e′ ∈ Bn} is bounded in
+CVP (∂)(Ω) for each n ∈ N, then there is a unique extension F ∈ CVP (∂)(Ω,E) of f.
+Proof. CVP (∂)(Ω) is a Fréchet–Schwartz space and (idEΩ,idKΩ) a strong,
+consistent generator for (CVP (∂),E) by Proposition 4.2.14 and the proof of Example
+4.2.16 b). Now, Theorem 5.2.40 and Proposition 5.2.8 prove our statement.
+□
+We already mentioned examples of families of weights V such that CVP (∂)(Rd)
+is a nuclear Fréchet space and sets of uniqueness for (idKRd ,CVP (∂)) in Remark
+4.2.20 and Remark 5.2.23 and if P(∂) = ∂ or P(∂) = ∆. Further sets of uniqueness
+are given in Remark 5.2.66. If E is a Banach space, then an almost norming set
+fixes the topology and examples can be found via Remark 5.2.35.
+Fν(Ω) a Banach space and E a Fréchet space. Let E be a Fréchet space,
+(Bn) fix the topology in E and recall the assumptions of Remark 5.2.26.
+Let
+(T E,T K) be a strong, consistent family for (F,E) and a generator for (Fν,E).
+Let F(Ω) and F(Ω,E) be ε-into-compatible and the inclusion Fν(Ω) ↪ F(Ω)
+continuous. Consider a set of uniqueness U for (T K,Fν) and G ∶= span(⋃n∈N Bn) ⊂
+
+5.2. EXTENSION OF VECTOR-VALUED FUNCTIONS
+91
+E′. For u ∈ F(Ω)εE such that u(B○F (Ω)′
+Fν(Ω) ) is bounded in E we have RU,G(f) ∈
+FνG(U,E) with f ∶= S(u) ∈ Fεν(Ω,E) by (50). We note that
+sup
+e′∈Bn
+∣fe′∣Fν(Ω) = sup
+e′∈Bn
+sup
+x∈ω ∣e′(T E(f)(x)ν(x))∣ = sup
+e′∈Bn
+sup
+y∈Nω(f)
+∣e′(y)∣
+with the bounded set Nω(f) ∶= {T E(f)(x)ν(x) ∣ x ∈ ω} ⊂ E, implying RU,G(f) ∈
+FVG(U,E)sb. Thus the injective linear map
+RU,G∶Fεν(Ω,E) → FνG(U,E)sb, f ↦ (T E(f)(x))x∈U,
+is well-defined.
+5.2.42. Question. Let the assumptions of Remark 5.2.26 be fulfilled, E be a
+Fréchet space, (Bn) fix the topology in E and G ∶= span(⋃n∈N Bn). When is the
+injective restriction map
+RU,G∶Fεν(Ω,E) → FνG(U,E)sb, f ↦ (T E(f)(x))x∈U,
+surjective?
+Now, we can generalise [70, Corollary 2.4, p. 692] and [93, Theorem 11, p. 5].
+5.2.43. Corollary. Let E be a Fréchet space, (Bn) fix the topology in E, set
+G ∶= span(⋃n∈N Bn) and let F(Ω) and F(Ω,E) be ε-into-compatible. Let (T E,T K)
+be a generator for (Fν,E) and a strong, consistent family for (F,E), Fν(Ω) a
+Banach space whose closed unit ball BFν(Ω) is a compact subset of F(Ω) and U a
+set of uniqueness for (T K,Fν). Then the restriction map
+RU,G∶Fεν(Ω,E) → FνG(U,E)sb
+is surjective.
+Proof. Let f ∈ FνG(U,E)sb.
+Then {fe′ ∣ e′ ∈ Bn} is bounded in Fν(Ω)
+for each n ∈ N. We deduce for each n ∈ N, (ak)k∈N ∈ ℓ1 and (e′
+k)k∈N ⊂ Bn that
+(∑k∈N ake′
+k)○f admits the extension ∑k∈N akfe′
+k in Fν(Ω). Due to [69, Proposition
+7, p. 503] the LB-space E′((Bn)n∈N) ∶= lim
+←�
+n∈N
+E′(Bn), where
+E′(Bn) ∶= {���
+k∈N
+ake′
+k ∣ (ak)k∈N ∈ ℓ1, (e′
+k)k∈N ⊂ Bn}
+is endowed with its Banach space topology for n ∈ N, determines boundedness in
+E.
+Hence we conclude that f ∈ FνE′((Bn)n∈N)(U,E), which yields that there is
+u ∈ F(Ω)εE with bounded u(B○F (Ω)′
+Fν(Ω) ) ⊂ E such that RU,G(S(u)) = f by Theorem
+5.2.29.
+□
+As an application we directly obtain the following two corollaries of Corol-
+lary 5.2.43 since its assumptions are fulfilled by the proof of Corollary 5.2.30 and
+Corollary 5.2.33, respectively.
+5.2.44. Corollary. Let E be a Fréchet space, (Bn) fix the topology in E and
+G ∶= span(⋃n∈N Bn), Ω ⊂ Rd open, P(∂)K a hypoelliptic linear partial differential
+operator, ν∶Ω → (0,∞) continuous and U a set of uniqueness for (idKΩ,CνP (∂)). If
+f∶U → E is a function such that e′ ○f admits an extension fe′ ∈ CνP (∂)(Ω) for each
+e′ ∈ G and {fe′ ∣ e′ ∈ Bn} is bounded in CνP (∂)(Ω) for each n ∈ N, then there exists
+a unique extension F ∈ CνP (∂)(Ω,E) of f.
+5.2.45. Corollary. Let E be a Fréchet space, (Bn) fix the topology in E and
+G ∶= span(⋃n∈N Bn), ν∶D → (0,∞) continuous and U∗ ⊂ D have an accumulation
+point in D. If f∶{0} ∪ ({1} × U∗) → E is a function such that there is fe′ ∈ Bν(D)
+for each e′ ∈ G with fe′(0) = e′(f(0)) and f ′
+e′(z) = e′(f(1,z)) for all z ∈ U∗ and
+
+92
+5. APPLICATIONS
+{fe′ ∣ e′ ∈ Bn} is bounded in Bν(D) for each n ∈ N, then there exists a unique
+F ∈ Bν(D,E) with F(0) = f(0) and (∂1
+C)EF(z) = f(1,z) for all z ∈ U∗.
+5.2.2. Extension from thick sets. In order to obtain an affirmative answer
+to Question 5.2.9 for general separating subspaces of E′ we have to restrict to the
+spaces FV(Ω) from Definition 3.1.4 and a certain class of sets of uniqueness.
+5.2.46. Definition (fix the topology). Let FV(Ω) be a dom-space. We say
+that U ⊂ ⋃m∈M({m}×ωm) fixes the topology in FV(Ω) if for every j ∈ J and m ∈ M
+there are i ∈ J, k ∈ M and C > 0 such that
+∣f∣j,m ≤ C
+sup
+x∈ωk
+(k,x)∈U
+∣T K
+k (f)(x)∣νi,k(x),
+f ∈ FV(Ω).
+In particular, U is a set of uniqueness if it fixes the topology. The present
+definition of fixing the topology is a generalisation of [30, Definition 13, p. 234].
+Sets that fix the topology appear under several different notions. Rubel and Shields
+call them dominating in [148, 4.10 Definition, p. 254] in the context of bounded
+holomorphic functions. In the context of the space of holomorphic functions with
+the topology of compact convergence studied by Grosse-Erdmann [81, p. 401] they
+are said to determine locally uniform convergence. Ehrenpreis [61, p. 3,4,13] (cf.
+[156, Definition 3.2, p. 166]) refers to them as sufficient sets when he considers
+inductive limits of weighted spaces of entire resp. holomorphic functions, including
+the case of Banach spaces. In the case of Banach spaces sufficient sets coincide
+with weakly sufficient sets defined by Schneider [156, Definition 2.1, p. 163] (see e.g.
+[102, §7, 1), p. 547]) and these notions are extended beyond spaces of holomorphic
+functions by Korobe˘ınik [102, p. 531]. Seip [162, p. 93] uses the term sampling sets
+in the context of weighted Banach spaces of holomorphic functions whereas Beurling
+uses the term balayage in [14, p. 341] and [14, Definition, p. 343]. Leibowitz [122,
+Exercise 4.1.4, p. 53], Stout [170, 7.1 Definition, p. 36] and Globevnik [76, p. 291–
+292] call them boundaries in the context of subalgebras of the algebra C(Ω,C) of
+complex-valued continuous functions on a compact Hausdorff space Ω with sup-
+norm. Fixing the topology is also connected to the notion of frames used by Bonet
+et al. in [31]. Let us set
+ℓV(U,E) ∶= {f∶U → E ∣ ∀ j ∈ J,m ∈ M,α ∈ A ∶ ∥f∥j,m,α < ∞}
+(53)
+with
+∥f∥j,m,α ∶=
+sup
+x∈ωm
+(m,x)∈U
+pα(f(m,x))νj,m(x)
+for an lcHs E and a set U which fixes the topology in FV(Ω). If M is a singleton,
+ωm = Ω = U, then ℓV(U,E) coincides with the space defined right above Example
+4.2.2. If U is countable, then the inclusion ℓV(U) ↪ KU continuous where KU is
+equipped with the topology of pointwise convergence and ℓV(U) contains the space
+of sequences (on U) with compact support as a linear subspace, then (T K
+k,x)(k,x)∈U
+is an ℓV(U)-frame in the sense of [31, Definition 2.1, p. 3].
+5.2.47. Definition (lb-restriction space). Let FV(Ω) be a dom-space, U fix
+the topology in FV(Ω) and G ⊂ E′ a separating subspace. We set
+NU,i,k(f) ∶= {f(k,x)νi,k(x) ∣ x ∈ ωk, (k,x) ∈ U}
+for i ∈ J, k ∈ M and f ∈ FVG(U,E) and
+FVG(U,E)lb ∶={f ∈ FVG(U,E) ∣ ∀ i ∈ J, k ∈ M ∶ NU,i,k(f) bounded in E}
+=FVG(U,E) ∩ ℓV(U,E).
+
+5.2. EXTENSION OF VECTOR-VALUED FUNCTIONS
+93
+Consider a set U which fixes the topology in FV(Ω), a separating subspace
+G ⊂ E′ and a strong, consistent family (T E
+m,T K
+m)m∈M for (FV,E). For u ∈ FV(Ω)εE
+set f ∶= S(u) ∈ FV(���,E) by Theorem 3.1.12. Then we have RU,G(f) ∈ FVG(U,E)
+with f ∶= S(u) by Remark 5.2.6 and for i ∈ J and k ∈ M
+sup
+y∈NU,i,k(RU,G(f))
+pα(y) =
+sup
+x∈ωk
+(k,x)∈U
+pα(T E
+k (f)(x))νi,k(x) ≤ ∣f∣i,k,α < ∞
+for all α ∈ A, implying the boundedness of NU,i,k(RU,G(f)) in E. Thus RU,G(f) ∈
+FVG(U,E)lb and the injective linear map
+RU,G∶S(FV(Ω)εE) → FVG(U,E)lb, f ↦ (T E
+m(f)(x))(m,x)∈U,
+is well-defined.
+5.2.48. Question. Let G ⊂ E′ be a separating subspace, (T E
+m,T K
+m)m∈M a
+strong, consistent generator for (FV,E) and U fix the topology in FV(Ω). When
+is the injective restriction map
+RU,G∶S(FV(Ω)εE) → FVG(U,E)lb, f ↦ (T E
+m(f)(x))(m,x)∈U,
+surjective?
+If G ⊂ E′ determines boundedness and U fixes the topology in FV(Ω), then
+the preceding question and Question 5.2.9 coincide.
+5.2.49. Remark. Let G ⊂ E′ determine boundedness, (T E
+m,T K
+m)m∈M a strong,
+consistent generator for (FV,E) and U fix the topology in FV(Ω). Then
+FVG(U,E)lb = FVG(U,E).
+Proof. We only need to show that the inclusion ‘⊃’ holds. Let f ∈ FVG(U,E).
+Then there is fe′ ∈ FV(Ω) with T K
+m(fe′)(x) = (e′ ○ f)(m,x) for all (m,x) ∈ U and
+sup
+y∈NU,i,k(f)
+∣e′(y)∣ =
+sup
+x∈ωk
+(k,x)∈U
+∣(e′ ○ f)(k,x)∣νi,k(x) ≤ ∣fe′∣i,k < ∞
+for each e′ ∈ G, i ∈ J and k ∈ M. Since G ⊂ E′ determines boundedness, this means
+that NU,i,k(f) is bounded in E and hence f ∈ FVG(U,E)lb.
+□
+FV(Ω) arbitrary and E a semi-Montel space.
+5.2.50. Definition (generalised Schwartz space). We call an lcHs E a gener-
+alised Schwartz space if every bounded set in E is already precompact.
+In particular, semi-Montel spaces and Schwartz spaces are generalised Schwartz
+spaces by [89, 10.4.3 Corollary, p. 202]. Conversely, a generalised Schwartz space is
+a Schwartz space if it is quasi-normable by [89, 10.7.3 Corollary, p. 215]. Moreover,
+looking at the proof of Lemma 3.2.2 b), we see that this lemma not only holds for
+semi-Montel or Schwartz spaces but for all generalised Schwartz spaces.
+5.2.51. Proposition. Let E be an lcHs, FV(Ω) a dom-space and U fix the
+topology in FV(Ω). Then Rf ∈ L(E′
+b,FV(Ω)) and Rf(B○
+α) is bounded in FV(Ω)
+for every f ∈ FVE′(U,E)lb and α ∈ A where Bα ∶= {x ∈ E ∣ pα(x) < 1} and Rf is
+the map from Remark 5.2.5. In addition, if E is a generalised Schwartz space, then
+Rf ∈ L(E′
+γ,FV(Ω)) and Rf(B○
+α) is relatively compact in FV(Ω).
+Proof. Let f ∈ FVE′(U,E)lb, j ∈ J and m ∈ M. Then there are i ∈ J, k ∈ M
+and C > 0 such that for every e′ ∈ E′
+∣Rf(e′)∣j,m = ∣fe′∣j,m ≤ C
+sup
+x∈ωk
+(k,x)∈U
+∣T K
+k (fe′)(x)∣νi,k(x)
+
+94
+5. APPLICATIONS
+= C
+sup
+x∈ωk
+(k,x)∈U
+∣(e′ ○ f)(k,x)∣νi,k(x) = C
+sup
+y∈NU,i,k(f)
+∣e′(y)∣,
+which proves the first part because NU,i,k(f) is bounded in E. Let us consider the
+second part. The bounded set NU,i,k(f) is already precompact in E because E is
+a generalised Schwartz space. Therefore we have Rf ∈ L(E′
+γ,FV(Ω)). The polar
+B○
+α is relatively compact in E′
+γ for every α ∈ A by the Alaoğlu–Bourbaki theorem
+and thus Rf(B○
+α) in FV(Ω) as well.
+□
+5.2.52. Theorem. Let E be a semi-Montel space, (T E
+m,T K
+m)m∈M a strong, con-
+sistent generator for (FV,E) and U fix the topology in FV(Ω). Then the restriction
+map RU,E′∶S(FV(Ω)εE) → FVE′(U,E)lb is surjective.
+Proof. Let f ∈ FVE′(Ω,E)lb and e′ ∈ E′. For every f ′ ∈ FV(Ω)′ there are
+j ∈ J, m ∈ M and C0 > 0 with
+∣Rt
+f(f ′)(e′)∣ = ∣f ′(fe′)∣ ≤ C0∣fe′∣j,m.
+By the proof of Proposition 5.2.51 there are i ∈ J, k ∈ M and C > 0 such that
+∣Rt
+f(f ′)(e′)∣ ≤ C0C
+sup
+y∈NU,i,k(f)
+∣e′(y)∣ ≤ C0C
+sup
+y∈acx(NU,i,k(f))
+∣e′(y)∣.
+The set acx(NU,i,k(f)) is absolutely convex and compact by [89, 6.2.1 Proposi-
+tion, p. 103] and [89, 6.7.1 Proposition, p. 112] because E is a semi-Montel space.
+Therefore Rt
+f(f ′) ∈ (E′
+κ)′ = J (E) by the Mackey–Arens theorem. As in Theorem
+5.2.15 we obtain J −1○Rt
+f ∈ FV(Ω)εE by (45), (46) and Proposition 5.2.51. Setting
+F ∶= S(J −1 ○ Rt
+f), we conclude T E
+m(F)(x) = f(m,x) for all (m,x) ∈ U by (47) and
+so RU,E′(F) = f.
+□
+5.2.53. Remark. Let E be a Fréchet space with increasing system of seminorms
+(pαn)n∈N, Bn ∶= B○
+αn where Bαn ∶= {x ∈ E ∣ pαn(x) < 1}, (T E
+m,T K
+m)m∈M a strong,
+consistent generator for (FV,E) and U a set of uniqueness for (T K
+m,FV)m∈M. If U
+fixes the topology of FV(Ω), then FVE′(U,E)sb = FVE′(U,E) by Remark 5.2.49
+and Proposition 5.2.51. Hence Theorem 5.2.52 answers Question 5.2.37 if E is a
+Fréchet–Montel space.
+Our first application of Theorem 5.2.52 concerns the space Cbu(Ω,E) of bounded
+uniformly continuous functions from a metric space Ω to an lcHs E from Example
+4.2.7.
+5.2.54. Corollary. Let Ω be a metric space, U ⊂ Ω a dense subset and E a
+semi-Montel space. If f∶U → E is a function such that e′ ○ f admits an extension
+fe′ ∈ Cbu(Ω) for each e′ ∈ E′, then there is a unique extension F ∈ Cbu(Ω,E) of f.
+In particular,
+Cbu(Ω,E) = {f∶Ω → E ∣ ∀ e′ ∈ E′ ∶ e′ ○ f ∈ Cbu(Ω)}.
+Proof. (idEΩ,idKΩ) is a strong, consistent generator for (Cbu,E) and we have
+Cbu(Ω)εE ≅ Cbu(Ω,E) via S by Example 4.2.7. Due to Theorem 5.2.52, Proposition
+5.2.8 and Remark 5.2.49 with G = E′ the extension F exists and is unique because
+the dense set U ⊂ Ω fixes the topology in Cbu(Ω). The rest follows from Proposition
+5.2.10.
+□
+Next, we consider the space A(Ω,E) of continuous functions from Ω to an lcHs
+E over C which are holomorphic on an open and bounded set Ω ⊂ C from Example
+4.2.13.
+
+5.2. EXTENSION OF VECTOR-VALUED FUNCTIONS
+95
+5.2.55. Corollary. Let Ω ⊂ C be open and bounded, U ⊂ Ω fix the topology in
+A(Ω) and E a semi-Montel space over C. If f∶U → E is a function such that e′ ○f
+admits an extension fe′ ∈ A(Ω) for each e′ ∈ E′, then there is a unique extension
+F ∈ A(Ω,E) of f. In particular,
+A(Ω,E) = {f∶Ω → E ∣ ∀ e′ ∈ E′ ∶ e′ ○ f ∈ A(Ω)}.
+Proof. (idEΩ,idCΩ) is a strong, consistent generator for (A,E) and A(Ω)εE ≅
+A(Ω,E) via S by Example 4.2.13. Due to Theorem 5.2.52, Proposition 5.2.8 and
+Remark 5.2.49 with G = E′ the extension F exists and is unique. The remaining
+part follows from Proposition 5.2.10.
+□
+If Ω ⊂ C is connected, then the boundary ∂Ω of Ω fixes the topology in A(Ω)
+by the maximum principle. If Ω = D, then ∂D is the intersection of all sets that fix
+the topology in A(D) by [170, 7.7 Example, p. 39].
+If E is a generalised Schwartz space which is not a semi-Montel space, we do not
+know whether the extension results in Corollary 5.2.54 and Corollary 5.2.55 hold
+but we still have a weak-strong principle due to the following observation which is
+based on [87, Chap. 3, §9, Proposition 2, p. 231] with σ(E,E′) replaced by σ(E,G).
+5.2.56. Proposition. If
+(i) E is a semi-Montel space and G ⊂ E′ a separating subspace, or
+(ii) E is a generalised Schwartz space and G ⊂ ̂E′ a separating subspace, i.e.
+separates the points of the completion ̂E,
+then the initial topology of E and the topology σ(E,G) coincide on the bounded sets
+of E.
+Proof. (i) Let B ⊂ E be a bounded set. If E is a semi-Montel space, then the
+closure B is compact in E. The topology induced by σ(E,G) on B is Hausdorff and
+weaker than the initial topology induced by E. Thus the two topologies coincide
+on B and so on B by the remarks above [87, Chap. 3, §9, Proposition 2, p. 231].
+(ii) Let B ⊂ E be a bounded set. If E is a generalised Schwartz space, then B is
+precompact in E and relatively compact in the completion ̂E by [89, 3.5.1 Theorem,
+p. 64]. Hence the closure B is compact in ̂E. The topology induced by σ( ̂E,G) on
+B is Hausdorff and weaker than the initial topology induced by ̂E, implying that
+the two topologies coincide on B as in part (i). This yields that σ(E,G) and the
+initial topology of E coincide on B because σ(E,G) = σ( ̂E,G) on B and the initial
+topologies of E and ̂E coincide on B as well.
+□
+Concerning (ii), we note that a separating subspace G ⊂ E′ of E need not
+separate the points of ̂E by [79, 5.4 Example, p. 36] (even though E′ = ̂E′ by [89,
+3.4.2 Theorem, p. 61–62]). Next, we apply Proposition 5.2.56 to the space A(Ω,E).
+5.2.57. Remark. Let E be an lcHs over C and Ω ⊂ C open and bounded. If
+(i) E is a semi-Montel space and G ⊂ E′ determines boundedness, or
+(ii) E is a generalised Schwartz space and G ⊂ ̂E′ a separating subspace which
+determines boundedness in E,
+then
+A(Ω,E) = {f∶Ω → E ∣ ∀ e′ ∈ G ∶ e′ ○ f ∈ A(Ω)}.
+Indeed, let us denote the right-hand side by A(Ω,E)σ and set Eσ ∶= (E,σ(E,G)).
+Then A(Ω,E)σ = A(Ω,Eσ) and f(Ω) is bounded for every f ∈ A(Ω,E)σ as G
+determines boundedness in E. The initial topology of E and σ(E,G) coincide on
+the bounded range f(Ω) of f ∈ A(Ω,E)σ by Proposition 5.2.56. Hence we deduce
+that
+A(Ω,E)σ = A(Ω,Eσ) = A(Ω,E).
+
+96
+5. APPLICATIONS
+In this way Bierstedt proves his weak-strong principles for weighted continuous
+functions in [17, 2.10 Lemma, p. 140] with G = E′ = ̂E′.
+FV(Ω) a Fréchet–Schwartz space and E locally complete.
+5.2.58. Definition (chain-structured). Let FV(Ω) be a dom-space. We say
+that U ⊂ ⋃m∈N({m} × ωm) is chain-structured if
+(i) (k,x) ∈ U
+⇒ ∀ m ∈ N, m ≥ k ∶ (m,x) ∈ U,
+(ii) ∀ (k,x) ∈ U, m ∈ N, m ≥ k, f ∈ FV(Ω) ∶ T K
+k (f)(x) = T K
+m(f)(x).
+5.2.59. Remark. Let Ω ⊂ Rd be open and V∞ a directed family of weights.
+Concerning the operators (T K
+m)m∈N0 of CV∞(Ω) from Example 3.1.9 a) where ωm ∶=
+{β ∈ Nd
+0 ∣ ∣β∣ ≤ m} × Ω resp. ωm ∶= Nd
+0 × Ω, we have for all k ∈ N0 and f ∈ CV∞(Ω)
+that
+T K
+k (f)(β,x) = ∂βf(x) = T K
+m(f)(β,x),
+β ∈ Nd
+0, ∣β∣ ≤ k, x ∈ Ω,
+for all m ∈ N0, m ≥ k. Hence condition (ii) of Definition 5.2.58 is fulfilled for any
+U ⊂ ⋃m∈N0({m} × ωm) in this case. Condition (i) says that once a ‘link’ (k,β,x)
+belongs to U for some order k, then the ‘link’ (m,β,x) belongs to U for any higher
+order m as well.
+5.2.60. Definition (diagonally dominated, increasing). We say that a family
+V ∶= (νj,m)j,m∈N of weights on Ω is diagonally dominated and increasing if ωm ⊂
+ωm+1 for all m ∈ N and νj,m ≤ νmax(j,m),max(j,m) on ωmin(j,m) for all j,m ∈ N as well
+as νj,j ≤ νj+1,j+1 on ωj for all j ∈ N.
+5.2.61. Remark. Let FV(Ω) be a dom-space, U ⊂ ⋃m∈N({m} × ωm) chain-
+structured, G ⊂ E′ a separating subspace and V diagonally dominated and increas-
+ing.
+a) If U fixes the topology in FV(Ω), then
+FVG(U,E)lb = {f ∈ FVG(U,E) ∣ ∀ i ∈ N ∶ NU,i(f) bounded in E}
+with NU,i(f) ∶= NU,i,i(f).
+b) Let FV(Ω) be a Fréchet space. We set Um ∶= {(m,x) ∈ U ∣ x ∈ ωm} and
+Bj ∶= ⋃j
+m=1{T K
+m,x(⋅)νm,m(x) ∣ (m,x) ∈ Um} ⊂ FV(Ω)′ for j ∈ N. Then
+U fixes the topology in FV(Ω) in the sense of Definition 5.2.46 if and
+only if the sequence (Bj)j∈N fixes the topology in FV(Ω) in the sense of
+Definition 5.2.34.
+Proof. Let us begin with a). We only need to show that the inclusion ‘⊃’ holds.
+Let f be an element of the right-hand side and i,k ∈ N. We set m ∶= max(i,k) and
+observe that for (k,x) ∈ U we have (m,x) ∈ U by (i) and
+(e′ ○ f)(k,x) = T K
+k (fe′)(x) =
+(ii) T K
+m(fe′)(x) = (e′ ○ f)(m,x)
+for each e′ ∈ G with (i) and (ii) from the definition of U being chain-structured.
+Since G is separating, it follows that f(k,x) = f(m,x). Hence we get for all α ∈ A
+sup
+y∈NU,i,k(f)
+pα(y) =
+sup
+x∈ωk
+(k,x)∈U
+pα(f(k,x))νi,k(x) ≤
+(i)
+sup
+x∈ωm
+(m,x)∈U
+pα(f(k,x))νm,m(x)
+=
+sup
+x∈ωm
+(m,x)∈U
+pα(f(m,x))νm,m(x) < ∞
+using that ωk ⊂ ωm and V is diagonally dominated.
+Let us turn to part b). ‘⇒’: Let j ∈ N and A ⊂ FV(Ω) be bounded. Then
+sup
+y∈Bj
+sup
+f∈A
+∣y(f)∣ =
+sup
+1≤m≤j
+(m,x)∈Um
+sup
+f∈A
+∣T K
+m(f)(x)∣νm,m(x) ≤ sup
+f∈A
+sup
+1≤m≤j
+∣f∣m,m < ∞
+
+5.2. EXTENSION OF VECTOR-VALUED FUNCTIONS
+97
+since A is bounded, implying that Bj is bounded in FV(Ω)′
+b. Further, (Bj) is
+increasing by definition. Additionally, for all j ∈ N
+B○
+j =
+j
+⋂
+m=1
+{f ∈ FV(Ω) ∣
+sup
+x∈ωm
+(m,x)∈U
+∣T K
+m(f)(x)∣νm,m(x) ≤ 1}
+= {f ∈ FV(Ω) ∣
+sup
+x∈ωj
+(j,x)∈U
+∣T K
+j (f)(x)∣νj,j(x) ≤ 1}
+because U is chain-structured and V increasing. Thus (B○
+j) is a fundamental system
+of zero neighbourhoods of FV(Ω) if U fixes the topology.
+‘⇐’: Let j,m ∈ N. Then there are i ∈ N and ε > 0 such that
+εB○
+i ⊂ {f ∈ FV(Ω) ∣ ∣f∣j,m ≤ 1} =∶ Dj,m
+which follows from fixing the topology in the sense of Definition 5.2.34. Let f ∈ Dj,m
+and set
+∣f∣Ui ∶=
+sup
+(i,x)∈Ui
+∣T K
+i (f)(x)∣νi,i(x).
+If ∣f∣Ui = 0, then tf ∈ εB○
+i for all t > 0 and hence t∣f∣j,m = ∣tf∣j,m ≤ 1 for all t > 0,
+which yields ∣f∣j,m = 0 = ∣f∣Ui. If ∣f∣Ui ≠ 0, then
+f
+∣f∣Ui ∈ B○
+i and thus ε
+f
+∣f∣Ui ∈ Dj,m,
+implying
+∣f∣j,m = 1
+ε∣f∣Ui∣ε f
+∣f∣Ui
+∣j,m ≤ 1
+ε∣f∣Ui.
+The inequality ∣f∣j,m ≤ 1
+ε∣f∣Ui still holds if ∣f∣Ui = 0.
+□
+5.2.62. Theorem ([30, Theorem 16, p. 236]). Let Y be a Fréchet–Schwartz
+space, (Bj)j∈N fix the topology in Y and A∶X ∶= span(⋃j∈N Bj) → E be a linear map
+which is bounded on each Bj. If
+a) (At)−1(Y ) is dense in E′
+b and E locally complete, or
+b) (At)−1(Y ) is dense in E′
+σ and E is Br-complete,
+then A has a (unique) extension ̂A ∈ Y εE.
+Now, we generalise [30, Theorem 17, p. 237].
+5.2.63. Theorem. Let E be an lcHs and G ⊂ E′ a separating subspace. Let
+(T E
+m,T K
+m)m∈M be a strong, consistent generator for (FV,E), FV(Ω) a Fréchet–
+Schwartz space, V diagonally dominated and increasing and U be chain-structured
+and fix the topology in FV(Ω). If
+a) G is dense in E′
+b and E locally complete, or
+b) E is Br-complete,
+then the restriction map RU,G∶S(FV(Ω)εE) → FVG(U,E)lb is surjective.
+Proof. Let f ∈ FVG(U,E)lb. We set X ∶= span(⋃j∈N Bj) with Bj from Re-
+mark 5.2.61 b) and Y ∶= FV(Ω). Let A∶X → E be the linear map determined
+by
+A(T K
+m,x(⋅)νm,m(x)) ∶= f(m,x)νm,m(x)
+for 1 ≤ m ≤ j and (m,x) ∈ Um with Um from Remark 5.2.61 b). The map A is
+well-defined since G is σ(E′,E)-dense, and bounded on each Bj because A(Bj) =
+⋃j
+m=1 NU,m(f).
+Let e′ ∈ G and fe′ be the unique element in FV(Ω) such that
+T K
+m(fe′)(x) = (e′ ○ f)(m,x) for all (m,x) ∈ U, which implies T K
+m(fe′)(x)νm,m(x) =
+(e′○A)(T K
+m,x(⋅)νm,m(x)) for all (m,x) ∈ Um. This equation allows us to consider fe′
+as a linear form on X (by fe′(T K
+m,x(⋅)��m,m(x)) ∶= (e′ ○ A)(T K
+m,x(⋅)νm,m(x))), which
+yields e′ ○ A ∈ FV(Ω) for all e′ ∈ G. It follows that G ⊂ (At)−1(Y ). Noting that G
+is σ(E′,E)-dense, we apply Theorem 5.2.62 and obtain an extension ̂A ∈ FV(Ω)εE
+
+98
+5. APPLICATIONS
+of A. We set F ∶= S(̂A) and observe that for all (m,x) ∈ U there is j ∈ N, j ≥ m,
+such that (j,x) ∈ Uj and νj,j(x) > 0 by (6) and because U is chain-structured and
+V diagonally dominated and increasing. Due to the proof of Remark 5.2.61 a) we
+have f(j,x) = f(m,x) and thus
+T E
+m(F)(x) = T E
+mS(̂A)(x) = ̂A(T K
+m,x) =
+1
+νj,j(x)
+̂A(T K
+m,x(⋅)νj,j(x))
+=
+1
+νj,j(x)
+̂A(T K
+j,x(⋅)νj,j(x)) = f(j,x) = f(m,x)
+by consistency, yielding RU,G(F) = f.
+□
+In particular, condition a) is fulfilled if E is semi-reflexive. Indeed, if E is semi-
+reflexive, then E is quasi-complete by [153, Chap. IV, 5.5, Corollary 1, p. 144] and
+G
+b(E′,E) = G
+τ(E′,E) = E′ by [89, 11.4.1 Proposition, p. 227] and the bipolar theorem.
+For instance, condition b) is satisfied if E is a Fréchet space or E = (C∞
+∂,b(D),β)
+which is a Br-complete space by Proposition 5.2.16 and is not a Fréchet space by
+Remark 4.2.23.
+As stated, our preceding theorem generalises [30, Theorem 17, p. 237] where
+FV(Ω) is a closed subspace of CW∞(Ω) for open, connected Ω ⊂ Rd. A characteri-
+sation of sets that fix the topology in the space CW∞
+∂ (Ω) of holomorphic functions
+on an open, connected set Ω ⊂ C is given in [30, Remark 14, p. 235]. The characteri-
+sation given in [30, Remark 14 (b), p. 235] is still valid and applied in [30, Corollary
+18, p. 238] for closed subspaces of CW∞
+P (∂)(Ω) where P(∂)K is a hypoelliptic linear
+partial differential operator which satisfies the maximum principle, namely, that
+U ⊂ Ω fixes the topology if and only if there is a sequence (Ωn)n∈N of relatively
+compact, open subsets of Ω with ⋃n∈N Ωn = Ω such that ∂Ωn ⊂ U ∩ Ωn+1 for all
+n ∈ N. Among the hypoelliptic operators P(∂)K satisfying the maximum principle
+are the Cauchy–Riemann operator ∂ and the Laplacian ∆. Further examples can
+be found in [74, Corollary 3.2, p. 33]. The statement of [30, Corollary 18, p. 238]
+for the space of holomorphic functions is itself a generalisation of [81, Theorem 2,
+p. 401] with [81, Remark 2 (a), p. 406] where E is Br-complete and of [92, Theo-
+rem 6, p. 10] where E is semi-reflexive. The case that G is dense in E′
+b and E is
+sequentially complete is covered by [77, 3.3 Satz, p. 228–229], not only for spaces
+of holomorphic functions, but for several classes of function spaces.
+Let us turn to other families of weights than W∞. Due to Proposition 4.2.19
+we already know that U ∶= {0} × C fixes the topology in CV∞
+∂ (C) = CV∂(C) and
+U ∶= {0} × Rd in CV∞
+∆(Rd) = CV∆(Rd) if V ∶= (νj)j∈N fulfils Condition 4.2.18 and
+V∞ ∶= (νj,m)j∈N,m∈N0 where νj,m∶{β ∈ Nd
+0 ∣ ∣β∣ ≤ m} × Rd → [0,∞), νj,m(β,x) ∶=
+νj(x).
+5.2.64. Corollary. Let E be an lcHs, G ⊂ E′ a separating subspace, V ∶=
+(νj)j∈N an increasing family of weights which is locally bounded away from zero
+on an open set Ω ⊂ Rd, P(∂)K a hypoelliptic linear partial differential operator,
+CVP (∂)(Ω) a Schwartz space and U ⊂ Ω fix the topology of CVP (∂)(Ω). If
+a) G is dense in E′
+b and E locally complete, or
+b) E is Br-complete,
+and f∶U → E is a function in ℓV(U) such that e′ ○ f admits an extension fe′ ∈
+CVP (∂)(Ω) for each e′ ∈ G, then there is a unique extension F ∈ CVP (∂)(Ω,E) of f.
+Proof. The existence of F follows from Proposition 4.2.14, Example 4.2.16 b)
+and Theorem 5.2.63 with (T E
+m,T K
+m)m∈M ∶= (idEΩ,idKΩ). The uniqueness of F is a
+result of Proposition 5.2.8.
+□
+
+5.2. EXTENSION OF VECTOR-VALUED FUNCTIONS
+99
+We have the following sufficient conditions on a family of weights V which
+guarantee the existence of a countable set U ⊂ C that fixes the topology of CV∂(C)
+due to Abanin and Varziev [2].
+5.2.65. Proposition. Let V ∶= (νj)j∈N where νj(z) ∶= exp(ajµ(z)−ϕ(z)), z ∈ C,
+with some continuous, subharmonic function µ∶C → [0,∞), a continuous function
+ϕ∶C → R and a strictly increasing, positive sequence (aj)j∈N with a ∶= limj→∞ aj ∈
+(0,∞]. Let there be
+(i) s ≥ 0 and C > 0 such that ∣ϕ(z) − ϕ(ζ)∣ ≤ C and ∣µ(z) − µ(ζ)∣ ≤ C for all
+z,ζ ∈ C with ∣z − ζ∣ ≤ (1 + ∣z∣)−s,
+(ii) max(ϕ(z),µ(z)) ≤ ∣z∣q + C0 for some q,C0 > 0 and
+(iii) ln(∣z∣) = O(µ(z)) as ∣z∣ → ∞ if a = ∞, or ln(∣z∣) = o(µ(z)) as ∣z∣ → ∞ if
+0 < a < ∞.
+Let (λk)k∈N be the sequence of simple zeros of a function L ∈ C̃V∂(C) having no
+other zeros where ̃V ∶= (ν2
+j /νmj)j∈N for some sequence (mj)j∈N in N. Suppose that
+there are j0 ∈ N and a sequence of circles {z ∈ C ∣ ∣z∣ = Rm} with Rm ↗ ∞ such that
+∣L(z)∣νj0(z) ≥ Cm,
+m ∈ N, z ∈ C, ∣z∣ = Rm,
+for some Cm ↗ ∞ and
+∣L′(λk)∣νj0(λk) ≥ 1
+for all sufficiently large k ∈ N.
+Then CV∂(C) is a nuclear Fréchet space for all a ∈ (0,∞] and U ∶= (λk)k∈N fixes the
+topology of CV∂(C) if a = ∞. If µ is a radial function, i.e. µ(z) = µ(∣z∣), z ∈ C, with
+µ(2z) ∼ µ(z) as ∣z∣ → ∞, then U fixes the topology of CV∂(C) for all a ∈ (0,∞].
+Proof. First, we check that Condition 4.2.18 is satisfied, which implies that
+CV∂(C) is a nuclear Fréchet space by Proposition 4.2.19. We set k ∶= max(s,2) and
+observe that (i) is also fulfilled with k instead of s. Let z ∈ C and ∥ζ∥∞,∥η∥∞ ≤
+(1/
+√
+2)(1 + ∣z∣)−k =∶ r(z). From ∣ ⋅ ∣ ≤
+√
+2∥ ⋅ ∥∞ and (i) it follows
+∣µ(z + ζ) − µ(z + η)∣ ≤ ∣µ(z + ζ) − µ(z)∣ + ∣µ(z) − µ(z + η)∣ ≤ 2C
+and thus µ(z+ζ) ≤ 2C+µ(z+η). In the same way we obtain −ϕ(z+ζ) ≤ 2C−ϕ(z+η).
+Hence we have
+ajµ(z + ζ) − ϕ(z + ζ) ≤ 2C(aj + 1) + ajµ(z + η) − ϕ(z + η)
+for j ∈ N, implying
+νj(z + ζ) ≤ e2C(aj+1)νj(z + η),
+which means that (α.1) of Condition 4.2.18 holds. By (iii) there are ε > 0 and
+R > 0 such that ln(∣z∣) ≤ εµ(z) for all z ∈ C with ∣z∣ ≥ R if a = ∞. This yields for all
+∣z∣ ≥ max(2,R) that
+ajµ(z) + k ln(1 + ∣z∣) ≤ ajµ(z) + k ln(∣z∣2) = ajµ(z) + 2k ln(∣z∣) ≤ ajµ(z) + 2kεµ(z).
+Since a = ∞, there is n ∈ N such that an ≥ aj + 2kε, resulting in
+ajµ(z) + k ln(1 + ∣z∣) ≤ anµ(z)
+for all ∣z∣ ≥ max(2,R). Therefore we derive
+ajµ(z) + k ln(1 + ∣z∣) ≤ anµ(z) + k ln(1 + max(2,R))
+(54)
+for all z ∈ C, which means that (α.2) and (α.3) hold with ψj(z) ∶= r(z). If 0 < a < ∞,
+for every ε > 0 there is R > 0 such that ln(∣z∣) ≤ εµ(z) for all z ∈ C with ∣z∣ ≥ R by
+(iii). Thus we may choose ε > 0 such that aj+1 −aj ≥ 2kε > 0 because (aj) is strictly
+increasing. We deduce that (54) with n ∶= j + 1 holds in this case as well and (α.2)
+and (α.3), too.
+Observing that the condition that U = (λk)k∈N is the sequence of simple zeros
+of a function L ∈ C̃V∂(C) means that L ∈ L (Φa
+ϕ,µ;U) and (i) that ϕ and µ vary
+
+100
+5. APPLICATIONS
+slowly w.r.t. r(z) ∶= (1 + ∣z∣)−s in the notation of [2, Definition, p. 579, 584] and [2,
+p. 585], respectively, the statement that U fixes the topology is a consequence of
+[2, Theorem 2, p. 585–586].
+□
+5.2.66. Remark.
+a) Let D ⊂ C be convex, bounded and open with 0 ∈ D.
+Let ϕ(z) ∶= HD(z) ∶= supζ∈D Re(zζ), z ∈ C, be the supporting function of
+D, µ(z) ∶= ln(1 + ∣z∣), z ∈ C, and aj ∶= j, j ∈ N. Then ϕ and µ fulfil the
+conditions of Proposition 5.2.65 with a = ∞ by [2, p. 586] and the existence
+of an entire function L which fulfils the conditions of Proposition 5.2.65
+is guaranteed by [3, Theorem 1.6, p. 1537]. Thus there is a countable
+set U ∶= (λk)k∈N ⊂ C which fixes the topology in A−∞
+D
+∶= CV∂(C) with
+V ∶= (exp(ajµ − ϕ))j∈N.
+b) An explicit construction of a set U ∶= (λk)k∈N ⊂ C which fixes the topology
+in A−∞
+D
+is given in [1, Algorithm 3.2, p. 3629]. This construction does not
+rely on the entire function L. In particular (see [31, p. 15]), for D ∶= D
+we have ϕ(z) = ∣z∣, for each k ∈ N we may take lk ∈ N, lk > 2πk2, and set
+λk,j ∶= krk,j, 1 ≤ j ≤ lk, where rk,j denote the lk-roots of unity. Ordering
+λk,j in a sequence of one index appropriately, we obtain a sequence which
+fixes the topology of A−∞
+D .
+c) Let µ∶C → [0,∞) be a continuous, subharmonic, radial function which
+increases with ∣z∣ and satisfies
+(i) supζ∈C,∥ζ∥∞≤r(z) µ(z + ζ) ≤ infζ∈C,∥ζ∥∞≤r(z) µ(z + ζ) + C for some con-
+tinuous function r∶C → (0,1] and C > 0,
+(ii) ln(1 + ∣z∣2) = o(µ(∣z∣)) as ∣z∣ → ∞,
+(iii) µ(2∣z∣) = O(µ(∣z∣)) as ∣z∣ → ∞.
+Then V ∶= (exp(−(1/j)µ))j∈N fulfils Condition 4.2.18 where (α.1) follows
+from (i) and (α.2), (α.3) as in the proof of Proposition 5.2.65.
+Thus
+CV∂(C) is a nuclear Fréchet space by Proposition 4.2.19. If µ(∣z∣) = o(∣z∣2)
+as ∣z∣ → ∞ or µ(∣z∣) = ∣z∣2, z ∈ C, then U ∶= {αn + iβm ∣ n,m ∈ Z} fixes the
+topology in the space A0
+µ ∶= CV∂(C) for any α,β > 0 by [31, Corollary 4.6,
+p. 20] and [31, Proposition 4.7, p. 20], respectively.
+d) For instance, the conditions on µ in c) are fulfilled for µ(z) ∶= ∣z∣γ, z ∈
+C, with 0 < γ ≤ 2 by [130, 1.5 Examples (3), p. 205].
+If γ = 1, then
+A0
+µ = A0
+∂(C) is the space of entire functions of exponential type zero (see
+Remark 4.2.20).
+e) More general characterisations of countable sets that fix the topology of
+CV∂(C) can be found in [2, Theorem 1, p. 580] and [31, Theorem 4.5, p.
+17].
+The spaces A0
+µ from c) are known as Hörmander algebras and the space A−∞
+D
+considered in a) is isomorphic to the strong dual of the Korenblum space A−∞(D)
+via Laplace transform by [132, Proposition 4, p. 580].
+Fν(Ω) a Banach space and E locally complete. For a dom-space Fν(Ω),
+a set U that fixes the topology in Fν(Ω) and a separating subspace G ⊂ E′ we have
+FνG(U,E)lb ={f ∈ FνG(U,E) ∣ NU(f) bounded in E}
+=FνG(U,E) ∩ ℓν(U,E)
+where NU(f) ∶= {f(x)ν(x) ∣ x ∈ U}. Let us recall the assumptions of Remark 5.2.26
+but now U fixes the topology. Let (T E,T K) be a strong, consistent family for (F,E)
+and a generator for (Fν,E). Let F(Ω) and F(Ω,E) be ε-into-compatible and the
+inclusion Fν(Ω) ↪ F(Ω) continuous. Consider a set U which fixes the topology in
+Fν(Ω) and a separating subspace G ⊂ E′. For u ∈ F(Ω)εE such that u(B○F (Ω)′
+Fν(Ω) ) is
+
+5.2. EXTENSION OF VECTOR-VALUED FUNCTIONS
+101
+bounded in E we have RU,G(f) ∈ FνG(U,E) with f ∶= S(u) ∈ Fεν(Ω,E) by (50).
+Further, T K
+x (⋅)ν(x) ∈ B○F (Ω)′
+Fν(Ω) for every x ∈ ω, which implies that
+sup
+x∈U
+pα(RU,G(f)(x))ν(x) = sup
+x∈U
+pα(u(T K
+x (⋅)ν(x))) ≤
+sup
+y′∈B○F (Ω)′
+Fν(Ω)
+pα(u(y′)) < ∞
+for all α ∈ A by consistency. Hence RU,G(f) ∈ FνG(U,E)lb. Therefore the injective
+linear map
+RU,G∶Fεν(Ω,E) → FνG(U,E)lb, f ↦ (T E(f)(x))x∈U,
+is well-defined and the question we want to answer is:
+5.2.67. Question. Let the assumptions of Remark 5.2.26 be fulfilled and U fix
+the topology in Fν(Ω). When is the injective restriction map
+RU,G∶Fεν(Ω,E) → FνG(U,E)lb, f ↦ (T E(f)(x))x∈U,
+surjective?
+5.2.68. Proposition ([70, Proposition 3.1, p. 692]). Let E be a locally complete
+lcHs, G ⊂ E′ a separating subspace and Z a Banach space whose closed unit ball
+BZ is a compact subset of an lcHs Y . Let B1 ⊂ B○Y ′
+Z
+such that B○Z
+1
+∶= {z ∈ Z ∣ ∀ y′ ∈
+B1 ∶ ∣y′(z)∣ ≤ 1} is bounded in Z. If A∶X ∶= spanB1 → E is a linear map which is
+bounded on B1 such that there is a σ(E′,E)-dense subspace G ⊂ E′ with e′ ○ A ∈ Z
+for all e′ ∈ G, then there exists a (unique) extension ̂A ∈ Y εE of A such that ̂A(B○Y ′
+Z
+)
+is bounded in E.
+The following theorem is a generalisation of [70, Theorem 3.2, p. 693] and [93,
+Theorem 12, p. 5].
+5.2.69. Theorem. Let E be a locally complete lcHs, G ⊂ E′ a separating sub-
+space and F(Ω) and F(Ω,E) be ε-into-compatible. Let (T E,T K) be a generator
+for (Fν,E) and a strong, consistent family for (F,E), Fν(Ω) a Banach space
+whose closed unit ball BFν(Ω) is a compact subset of F(Ω) and U fix the topology
+in Fν(Ω). Then the restriction map
+RU,G∶Fεν(Ω,E) → FνG(U,E)lb
+is surjective.
+Proof. Let f ∈ FνG(U,E)lb. We set B1 ∶= {T K
+x (⋅)ν(x) ∣ x ∈ U}, X ∶= spanB1,
+Y ∶= F(Ω) and Z ∶= Fν(Ω). We have B1 ⊂ Y ′ since (T E,T K) is a consistent family
+for (F,E). If f ∈ BZ, then
+∣T K
+x (f)ν(x)∣ ≤ ∣f∣Fν(Ω) ≤ 1
+for all x ∈ U and thus B1 ⊂ B○Y ′
+Z
+. Further on, there is C > 0 such that for all f ∈ B○Z
+1
+∣f∣Fν(Ω) ≤ C sup
+x∈U
+∣T K
+x (f)∣ν(x) ≤ C
+as U fixes the topology in Z, implying the boundedness of B○Z
+1
+in Z. Let A∶X → E
+be the linear map determined by
+A(T K
+x (⋅)ν(x)) ∶= f(x)ν(x).
+The map A is well-defined since G is σ(E′,E)-dense, and bounded on B1 be-
+cause A(B1) = NU(f).
+Let e′ ∈ G and fe′ be the unique element in Fν(Ω)
+such that T K(fe′)(x) = (e′ ○ f)(x) for all x ∈ U, which implies T K(fe′)(x)ν(x) =
+(e′○A)(T K
+x (⋅)ν(x)). Again, this equation allows us to consider fe′ as a linear form on
+X (by setting fe′(T K
+x (⋅)ν(x)) ∶= (e′○A)(T K
+x (⋅)ν(x))), which yields e′○A ∈ Fν(Ω) = Z
+for all e′ ∈ G.
+Hence we can apply Proposition 5.2.68 and obtain an extension
+
+102
+5. APPLICATIONS
+̂A ∈ Y εE of A such that ̂A(B○Y ′
+Z
+) is bounded in E. We set ̃F ∶= S(̂A) ∈ Fεν(Ω,E)
+and get for all x ∈ U that
+T E( ̃F)(x) = T ES(̂A)(x) = ̂A(T K
+x ) =
+1
+ν(x)A(T K
+x (⋅)ν(x)) = f(x)
+by consistency for (F,E), yielding RU,G( ̃F) = f.
+□
+5.2.70. Corollary. Let E be a locally complete lcHs, G ⊂ E′ a separating
+subspace, Ω ⊂ Rd open, P(∂)K a hypoelliptic linear partial differential operator,
+ν∶Ω → (0,∞) continuous and U fix the topology in CνP (∂)(Ω). If f∶U → E is a
+function in ℓν(U,E) such that e′ ○ f admits an extension fe′ ∈ CνP (∂)(Ω) for every
+e′ ∈ G, then there exists a unique extension F ∈ CνP (∂)(Ω,E) of f.
+Proof. Observing that f ∈ FνG(U,E)lb with Fν(Ω) = CνP (∂)(Ω), our state-
+ment follows directly from Theorem 5.2.69 whose conditions are fulfilled by the
+proof of Corollary 5.2.30.
+□
+Sets that fix the topology in CνP (∂)(Ω) for different weights ν are well-studied
+if P(∂) = ∂ is the Cauchy–Riemann operator. If Ω ⊂ C is open, P(∂) = ∂ and ν = 1,
+then CνP (∂)(Ω) = H∞(Ω) is the space of bounded holomorphic functions on Ω.
+Brown, Shields and Zeller characterise the countable discrete sets U ∶= (zn)n∈N ⊂ Ω
+that fix the topology in H∞(Ω) with C = 1 and equality in Definition 5.2.46 for
+Jordan domains Ω in [36, Theorem 3, p. 167]. In particular, they prove for Ω = D
+that a discrete U = (zn)n∈N fixes the topology in H∞(D) if and only if almost every
+boundary point is a non-tangential limit of a sequence in U. Bonsall obtains the
+same characterisation for bounded harmonic functions, i.e. P(∂) = ∆ and ν = 1, on
+Ω = D by [32, Theorem 2, p. 473]. An example of such a set U = (zn)n∈N ⊂ D is
+constructed in [36, Remark 6, p. 172]. Probably the first example of a countable
+discrete set U ⊂ D that fixes the topology in H∞(D) is given by Wolff in [183, p.
+1327] (cf. [81, Theorem (Wolff), p. 402]). In [148, 4.14 Theorem, p. 255] Rubel and
+Shields give a charaterisation of sets U ⊂ Ω that fix the topology in H∞(Ω) by means
+of bounded complex measures where Ω ⊂ C is open and connected. The existence
+of a countable U fixing the topology in H∞(Ω) is shown in [148, 4.15 Proposition,
+p. 256]. In the case of several complex variables the existence of such a countable
+U is treated by Sibony in [167, Remarques 4 b), p. 209] and by Massaneda and
+Thomas in [128, Theorem 2, p. 838].
+If Ω = C and P(∂) = ∂, then CνP (∂)(Ω) =∶ F ∞
+ν (C) is a generalised L∞-version
+of the Bargmann–Fock space. In the case that ν(z) = exp(−α∣z∣2/2), z ∈ C, for
+some α > 0, Seip and Wallstén show in [162, Theorem 2.3, p. 93] that a countable
+discrete set U ⊂ C fixes the topology in F ∞
+ν (C) if and only if U contains a uniformly
+discrete subset U ′ with lower uniform density D−(U ′) > α/π (the proof of sufficiency
+is given in [165] and the result was announced in [161, Theorem 1.3, p. 324]). A
+generalisation of this result using lower angular densities is given by Lyubarski˘ı and
+Seip in [124, Theorem 2.2, p. 162] to weights of the form ν(z) = exp(−φ(arg z)∣z∣2/2),
+z ∈ C, with a 2π-periodic 2-trigonometrically convex function φ such that φ ∈
+C2([0,2π]) and φ(θ) + (1/4)φ′′(θ) > 0 for all θ ∈ [0,2π]. An extension of the results
+in [162] to weights of the form ν(z) = exp(−φ(z)), z ∈ C, with a subharmonic
+function φ such that ∆φ(z) ∼ 1 is given in [135, Theorem 1, p. 249] by Ortega-
+Cerdà and Seip.
+Here, f(x) ∼ g(x) for two functions f,g∶Ω → R means that
+there are C1,C2 > 0 such that C1g(x) ≤ f(x) ≤ C2g(x) for all x ∈ Ω.
+Marco,
+Massaneda and Ortega-Cerdà describe sets that fix the topology in F ∞
+ν (C) with
+ν(z) = exp(−φ(z)), z ∈ C, for some subharmonic function φ whose Laplacian ∆φ is
+a doubling measure (see [126, Definition 5, p. 868]), e.g. φ(z) = ∣z∣β for some β > 0,
+
+5.2. EXTENSION OF VECTOR-VALUED FUNCTIONS
+103
+in [126, Theorem A, p. 865]. The case of several complex variables is handled by
+Ortega-Cerdà, Schuster and Varolin in [136, Theorem 2, p. 81].
+If Ω = D and P(∂) = ∂, then CνP (∂)(Ω) =∶ A∞
+ν (D) is a generalised L∞-version
+of the weighted Bergman space (and of H∞(D)). For ν(z) = (1 − ∣z∣2)n, z ∈ D, for
+some n ∈ N, Seip proves that a countable discrete set U ⊂ D fixes the topology in
+A∞
+ν (D) if and only if U contains a uniformly discrete subset U ′ with lower uniform
+density D−(U ′) > n by [163, Theorem 1.1, p. 23], and gives a typical example
+in [163, p. 23].
+Later on, this is extended by Seip in [164, Theorem 2, p. 718]
+to weights ν(z) = exp(−φ(z)), z ∈ D, with a subharmonic function φ such that
+∆φ(z) ∼ (1 − ∣z∣2)−2, e.g. φ(z) = −β ln(1 − ∣z∣2), z ∈ D, for some β > 0. Domański
+and Lindström give necessary resp. sufficient conditions for fixing the topology in
+A∞
+ν (D) in the case that ν is an essential weight on D, i.e. there is C > 0 with
+ν(z) ≤ ̃ν(z) ≤ Cν(z) for each z ∈ D where ̃ν(z) ∶= (sup{∣f(z)∣ ∣ f ∈ BA∞
+ν (D)})−1 is
+the associated weight. In [55, Theorem 29, p. 260] they describe necessary resp.
+sufficient conditions for fixing the topology if the upper index Uν is finite (see [55,
+p. 242]), and necessary and sufficient conditions in [55, Corollary 31, p. 261] if
+0 < Lν = Uν < ∞ holds where Lν is the lower index (see [55, p. 243]), which for
+example can be applied to ν(z) = (1 − ∣z∣2)n(ln(
+e
+1−∣z∣))β, z ∈ D, for some n > 0 and
+β ∈ R. The case of simply connected open Ω ⊂ C is considered in [55, Corollary 32,
+p. 261–262].
+Borichev, Dhuez and Kellay treat A∞
+ν (D) and F ∞
+ν (C) simultaneously.
+Let
+ΩR ∶= D, if R = 1, and ΩR ∶= C if R = ∞. They take ν(z) = exp(−φ(z)), z ∈ ΩR,
+where φ∶[0,R) → [0,∞) is an increasing function such that φ(0) = 0, limr→R φ(r) =
+∞, φ is extended to ΩR by φ(z) ∶= φ(∣z∣), φ ∈ C2(ΩR), and, in addition ∆φ(z) ≥ 1
+if R = ∞ (see [33, p. 564–565]). Then they set ρ∶[0,R) → R, ρ(r) ∶= [∆φ(r)]−1/2,
+and suppose that ρ decreases to 0 near R, ρ′(r) → 0, r → R, and either (ID) the
+function r ↦ ρ(r)(1 − r)−C increases for some C ∈ R and for r close to 1, resp. (IC)
+the function r ↦ ρ(r)rC increases for some C ∈ R and for large r, or (IIΩR) that
+ρ′(r)ln(1/ρ(r)) → 0, r → R (see [33, p. 567–569]). Typical examples for (ID) are
+φ(r) = ln(ln( 1
+1−r))ln( 1
+1−r)
+or
+φ(r) =
+1
+1−r,
+a typical example for (IID) is φ(r) = exp( 1
+1−r), for (IC)
+φ(r) = r2 ln(ln(r))
+or
+φ(r) = rp, for some p > 2,
+and a typical example for (IIC) is φ(r) = exp(r). Sets that fix the topology in
+A∞
+ν (D) are described by densities in [33, Theorem 2.1, p. 568] and sets that fix the
+topology in F ∞
+ν (C) in [33, Theorem 2.5, p. 569].
+Wolf uses sets that fix the topology in A∞
+ν (D) for the characterisation of
+weighted composition operators on A∞
+ν (D) with closed range in [182, Theorem
+1, p. 36] for bounded ν.
+5.2.71. Corollary. Let E be a locally complete lcHs, G ⊂ E′ a separating
+subspace, ν∶D → (0,∞) continuous and U ∶= {0} ∪ ({1} × U∗) fix the topology in
+Bν(D) with U∗ ⊂ D. If f∶U → E is a function in ℓν∗(U,E) such that there is fe′ ∈
+Bν(D) for each e′ ∈ G with fe′(0) = e′(f(0)) and f ′
+e′(z) = e′(f(1,z)) for all z ∈ U∗,
+then there exists a unique F ∈ Bν(D,E) with F(0) = f(0) and (∂1
+C)EF(z) = f(1,z)
+for all z ∈ U∗.
+Proof. As in Corollary 5.2.70 but with Fν∗(D) = Bν(D) and Corollary 5.2.33
+instead of Corollary 5.2.30.
+□
+Sets that fix the topology in Bν(D) play an important role in the characteri-
+sation of composition operators on Bν(D) with closed range. Chen and Gauthier
+give a characterisation in [42] for weights of the form ν(z) = (1 − ∣z∣2)α, z ∈ D,
+
+104
+5. APPLICATIONS
+for some α ≥ 1. We recall the following definitions which are needed to phrase
+this characterisation. For a continuous function ν∶D → (0,∞) and a non-constant
+holomorphic function φ∶D → D we set
+τ ν
+φ(z) ∶= ν(z)∣φ′(z)∣
+ν(φ(z)) , z ∈ D,
+and
+Ων
+ε ∶= {z ∈ D ∣ τ ν
+φ(z) ≥ ε}, ε > 0,
+and define the pseudohyperbolic distance
+ρ(z,w) ∶= ∣ z − w
+1 − zw∣, z,w ∈ D.
+For 0 < r < 1 a set B ⊂ D is called a pseudo r-net if for every w ∈ D there is z ∈ B
+with ρ(z,w) ≤ r (see [42, p. 195]). A set U⋆ ⊂ D is a sampling set for Bν(D) with ν
+as above in the sense of [42, p. 198] if and only if {0}∪({1}×U⋆) fixes the topology
+in Bν(D) (see the definitions above Corollary 5.2.33).
+5.2.72. Theorem ([42, Theorem 3.1, p. 199, Theorem 4.3, p. 202]). Let φ∶D →
+D be a non-constant holomorphic function and ν(z) = (1 − ∣z∣2)α, z ∈ D, for some
+α ≥ 1. Then the following statements are equivalent.
+(i) The composition operator Cφ∶Bν(D) → Bν(D), Cφ(f) ∶= f ○ φ, is bounded
+below (i.e. has closed range).
+(ii) There is ε > 0 such that {0} ∪ ({1} × φ(Ων
+ε)) fixes the topology in Bν(D).
+(iii) There are ε > 0 and 0 < r < 1 such that φ(Ων
+ε) is a pseudo r-net.
+This theorem has some predecessors. The implications (i)⇒(iii) and (iii), r <
+1/4 ⇒(i) for α = 1 are due to Ghatage, Yan and Zheng by [72, Proposition 1,
+p. 2040] and [72, Theorem 2, p. 2043]. This was improved by Chen to (i)⇔(iii)
+for α = 1 by removing the restriction r < 1/4 in [41, Theorem 1, p. 840].
+The
+proof of the equivalence (i)⇔(ii) given in [73, Theorem 1, p. 1372] for α = 1 is
+due to Ghatage, Zheng and Zorboska. A non-trivial example of a sampling set
+for α = 1 can be found in [73, Example 2, p. 1376] (cf. [42, p. 203]). In the case
+of several complex variables a characterisation corresponding to Theorem 5.2.72
+is given by Chen in [41, Theorem 2, p. 844] and Deng, Jiang and Ouyang in [50,
+Theorem 1–3, p. 1031–1032, 1034] where Ω is the unit ball of Cd. Giménez, Malavé
+and Ramos-Fernández extend Theorem 5.2.72 by [75, Theorem 3, p. 112] and [75,
+Corollary 1, p. 113] to more general weights of the form ν(z) = µ(1−∣z∣2) with some
+continuous function µ∶(0,1] → (0,∞) such that µ(r) → 0, r → 0+, which can be
+extended to a holomorphic function µ0 on D1(1) without zeros in D1(1) and fulfilling
+µ(1−∣1−z∣) ≤ C∣µ0(z)∣ for all z ∈ D1(1) and some C > 0 (see [75, p. 109]). Examples
+of such functions µ are µ1(r) ∶= rα, α > 0, µ2 ∶= r ln(2/r) and µ3(r) ∶= rβ ln(1 − r),
+β > 1, for r ∈ (0,1] (see [75, p. 110]) and with ν(z) = µ1(1 − ∣z∣2) = (1 − ∣z∣2)α, z ∈ D,
+one gets Theorem 5.2.72 back for α ≥ 1. For 0 < α < 1 and ν(z) = µ1(1−∣z∣2), z ∈ D,
+the equivalence (i)⇔(ii) is given in [184, Proposition 4.4, p. 14] of Yoneda as well and
+a sufficient condition implying (ii) in [184, Corollary 4.5, p. 15]. Ramos-Fernández
+generalises the results given in [75] to bounded essential weights ν on D by [144,
+Theorem 4.3, p. 85] and [144, Remark 4.2, p. 84]. In [141, Theorem 2.4, p. 3106]
+Pirasteh, Eghbali and Sanatpour use sets that fix the topology in Bν(D) for radial
+essential ν to characterise Li–Stević integral-type operators on Bν(D) with closed
+range instead of composition operators. The composition operator on the harmonic
+variant of the Bloch type space Bν(D) with ν(z) = (1 −∣z∣2)α, z ∈ D, for some α > 0
+is considered by Esmaeili, Estaremi and Ebadian, who give a corresponding result
+in [64, Theorem 2.8, p. 542].
+5.3. Weak-strong principles for differentiability of finite order
+This section is dedicated to Ck-weak-strong principles for differentiable func-
+tions. So the question is:
+
+5.3. WEAK-STRONG PRINCIPLES FOR DIFFERENTIABILITY OF FINITE ORDER
+105
+5.3.1. Question. Let E be an lcHs, G ⊂ E′ a separating subspace, Ω ⊂ Rd
+open and k ∈ N0 ∪ {∞}. If f∶Ω → E is such that e′ ○ f ∈ Ck(Ω) for each e′ ∈ G, does
+f ∈ Ck(Ω,E) hold?
+An affirmative answer to the preceding question is called a Ck-weak-strong
+principle. It is a result of Bierstedt [17, 2.10 Lemma, p. 140] that for k = 0 the
+C0-weak-strong principle holds if Ω ⊂ Rd is open (or more general a Hausdorff kR-
+space), G = E′ and E is such that every bounded set is already precompact in E,
+i.e. E is a generalised Schwartz space (see Definition 5.2.50 and Remark 5.2.57).
+For instance, the last condition is fulfilled if E is a semi-Montel or Schwartz space.
+The C0-weak-strong principle does not hold for general E by [94, Beispiel, p. 232].
+Grothendieck sketches in a footnote [82, p. 39] (cf. [84, Chap. 3, Sect. 8, Corol-
+lary 1, p. 134]) the proof that for k < ∞ a weakly-Ck+1 function f∶Ω → E on an
+open set Ω ⊂ Rd with values in a quasi-complete lcHs E is already Ck, i.e. that
+from e′ ○ f ∈ Ck+1(Ω) for all e′ ∈ E′ it follows f ∈ Ck(Ω,E). A detailed proof of this
+statement is given by Schwartz in [158], simultaneously weakening the condition
+from quasi-completeness of E to sequential completeness and from weakly-Ck+1 to
+weakly-Ck,1
+loc .
+5.3.2. Theorem ([158, Appendice, Lemme II, Remarques 10), p. 146–147]).
+Let E be a sequentially complete lcHs, Ω ⊂ Rd open and k ∈ N0.
+a) If f∶Ω → E is such that e′ ○ f ∈ Ck,1
+loc (Ω) for all e′ ∈ E′, then f ∈ Ck(Ω,E).
+b) If f∶Ω → E is such that e′ ○ f ∈ Ck+1(Ω) for all e′ ∈ E′, then f ∈ Ck(Ω,E).
+Here Ck,1
+loc (Ω) denotes the space of functions in Ck(Ω) whose partial derivatives
+of order k are locally Lipschitz continuous.
+Part b) clearly implies a C∞-weak-
+strong principle for open Ω ⊂ Rd, G = E′ and sequentially complete E. This can
+be generalised to locally complete E. Waelbroeck has shown in [177, Proposition
+2, p. 411] and [176, Definition 1, p. 393] that the C∞-weak-strong principle holds if
+Ω is a manifold, G = E′ and E is locally complete. It is a result of Bonet, Frerick
+and Jordá that the C∞-weak-strong principle still holds by [30, Theorem 9, p. 232]
+if Ω ⊂ Rd is open, G ⊂ E′ determines boundedness and E is locally complete. Due
+to [104, 2.14 Theorem, p. 20] of Kriegl and Michor an lcHs E is locally complete if
+and only if the C∞-weak-strong principle holds for Ω = R and G = E′.
+One of the goals of this section is to improve Theorem 5.3.2. We recall the
+following definition from Example 4.2.28. For k ∈ N0 the space of k-times con-
+tinuously partially differentiable E-valued functions on an open set Ω ⊂ Rd whose
+partial derivatives up to order k are continuously extendable to the boundary of Ω
+is
+Ck(Ω,E) = {f ∈ Ck(Ω,E) ∣ (∂β)Ef cont. extendable on Ω for all β ∈ Nd
+0, ∣β∣ ≤ k}
+equipped with the system of seminorms given by
+∣f∣Ck(Ω),α =
+sup
+x∈Ω
+β∈Nd
+0,∣β∣≤k
+pα((∂β)Ef(x)),
+f ∈ Ck(Ω,E), α ∈ A.
+The space of functions in Ck(Ω,E) such that all its k-th partial derivatives are
+γ-Hölder continuous with 0 < γ ≤ 1 is given by
+Ck,γ(Ω,E) ∶= {f ∈ Ck(Ω,E) ∣ ∀ α ∈ A ∶ ∣f∣Ck,γ(Ω),α < ∞}
+where
+∣f∣Ck,γ(Ω),α ∶= max(∣f∣Ck(Ω),α,
+sup
+β∈Nd
+0,∣β∣=k
+∣(∂β)Ef∣C0,γ(Ω),α)
+
+106
+5. APPLICATIONS
+with
+∣f∣C0,γ(Ω),α ∶= sup
+x,y∈Ω
+x≠y
+pα(f(x) − f(y))
+∣x − y∣γ
+.
+We set Ck,γ(Ω) ∶= Ck,γ(Ω,K) and
+ω1 ∶= {β ∈ Nd
+0 ∣ ∣β∣ ≤ k} × Ω
+and
+ω2 ∶= {β ∈ Nd
+0 ∣ ∣β∣ = k} × (Ω2 ∖ {(x,x) ∣ x ∈ Ω})
+as well as ω ∶= ω1 ∪ ω2. We define the operator T E∶Ck(Ω,E) → Eω by
+T E(f)(β,x) ∶=(∂β)E(f)(x)
+, (β,x) ∈ ω1,
+T E(f)(β,(x,y)) ∶=(∂β)E(f)(x) − (∂β)E(f)(y)
+, (β,(x,y)) ∈ ω2.
+and the weight ν∶ω → (0,∞) by
+ν(β,x) ∶= 1, (β,x) ∈ ω1,
+and
+ν(β,(x,y)) ∶=
+1
+∣x − y∣γ , (β,(x,y)) ∈ ω2.
+By setting F(Ω,E) ∶= Ck(Ω,E) and observing that
+∣f∣Ck,γ(Ω),α = sup
+x∈ω pα(T E(f)(x))ν(x),
+f ∈ Ck,γ(Ω,E), α ∈ A,
+we have Fν(Ω,E) = Ck,γ(Ω,E) with generator (T E,T K).
+5.3.3. Corollary. Let E be a locally complete lcHs, G ⊂ E′ determine bound-
+edness, Ω ⊂ Rd open and bounded, k ∈ N0 and 0 < γ ≤ 1. In the case k ≥ 1, assume
+additionally that Ω has Lipschitz boundary. If f∶Ω → E is such that e′ ○f ∈ Ck,γ(Ω)
+for all e′ ∈ G, then f ∈ Ck,γ(Ω,E).
+Proof. We take F(Ω) ∶= Ck(Ω) and F(Ω,E) ∶= Ck(Ω,E) and have Fν(Ω) =
+Ck,γ(Ω) and Fν(Ω,E) = Ck,γ(Ω,E) with the weight ν and generator (T E,T K) for
+(Fν,E) described above. Due to the proof of Example 4.2.28 and Theorem 3.1.12
+the spaces F(Ω) and F(Ω,E) are ε-into-compatible for any lcHs E (the condi-
+tion that E has metric ccp in Example 4.2.28 is only needed for ε-compatibility).
+Another consequence of Example 4.2.28 is that
+T E(S(u))(β,x) = (∂β)E(S(u))(x) = u(δx ○ (∂β)K) = u(T K
+β,x),
+(β,x) ∈ ω1,
+holds for all u ∈ F(Ω)εE, implying
+T E(S(u))(β,(x,y)) = T E(S(u))(β,x) − T E(S(u))(β,y) = u(T K
+β,x) − u(T K
+β,y)
+= u(T K
+β,(x,y)),
+(β,(x,y)) ∈ ω2.
+Thus (T E,T K) is a consistent family for (F,E) and its strength is easily seen. In
+addition, Fν(Ω) = Ck,γ(Ω) is a Banach space by [57, Theorem 9.8, p. 110] (cf. [4,
+1.7 Hölderstetige Funktionen, p. 46]) whose closed unit ball is compact in F(Ω) =
+Ck(Ω) by [4, 8.6 Einbettungssatz in Hölder-Räumen, p. 338]. Moreover, the ε-into-
+compatibility of F(Ω) and F(Ω,E) in combination with the consistency of (T E,T K)
+for (F,E) implies Fεν(Ω,E) ⊂ Fν(Ω,E) as linear spaces by Proposition 5.2.25
+c). Hence our statement follows from Theorem 5.2.29 with the set of uniqueness
+U ∶= {0} × Ω for (T K,Fν).
+□
+5.3.4. Remark. We point out that Corollary 5.3.3 corrects our result [117,
+Corollary 5.3, p. 16] by adding the missing assumption that Ω should additionally
+have Lipschitz boundary in the case k ≥ 1. This is needed to deduce that the closed
+unit ball of Ck,γ(Ω) is compact in Ck(Ω) by [4, 8.6 Einbettungssatz in Hölder-
+Räumen, p. 338] (in the notation of [74] Ω having Lipschitz boundary means that
+it is a C0,1 domain, see [74, Lemma 6.36, p. 136] and the comments below and
+above this lemma). This additional assumption is missing in [57, Theorem 14.32,
+p. 232], which is our main reference in [117] for the compact embedding, but it is
+
+5.3. WEAK-STRONG PRINCIPLES FOR DIFFERENTIABILITY OF FINITE ORDER
+107
+needed due to [4, U8.1 Gegenbeispiel zu Einbettungssätzen, p. 365] (cf. [74, p. 53]).
+However, this only affects the result [117, Corollary 6.3, p. 21–22] where we have to
+add this missing assumption as well (see Corollary 5.4.4 for this). The other results
+of [117] derived from [117, Corollary 5.3, p. 16] are not affected by this missing
+assumption since they are all a consequence of [117, Corollary 5.4, p. 17] and [117,
+Corollary 6.4, p. 22], whose proofs can be adjusted without additional assumptions
+(see Corollary 5.3.5 and Corollary 5.4.5 for this).
+Next, we use the preceding corollary to generalise the theorem of Grothendieck
+and Schwartz on weakly Ck+1-functions. For k ∈ N0 and 0 < γ ≤ 1 we define the
+space of k-times continuously partially differentiable E-valued functions with locally
+γ-Hölder continuous partial derivatives of k-th order on an open set Ω ⊂ Rd by
+Ck,γ
+loc (Ω,E) ∶= {f ∈ Ck(Ω,E) ∣ ∀ K ⊂ Ω compact, α ∈ A ∶ ∣f∣K,α < ∞}
+where
+∣f∣K,α ∶= max(∣f∣Ck(K),α,
+sup
+β∈Nd
+0,∣β∣=k
+∣(∂β)Ef∣C0,γ(K),α)
+with
+∣f∣Ck(K),α ∶=
+sup
+x∈K
+β∈Nd
+0,∣β∣≤k
+pα((∂β)Ef(x))
+and
+∣f∣C0,γ(K),α ∶= sup
+x,y∈K
+x≠y
+pα(f(x) − f(y))
+∣x − y∣γ
+.
+Further, we set Ck,γ
+loc (Ω) ∶= Ck,γ
+loc (Ω,K). Using Corollary 5.3.3, we are able to
+improve Theorem 5.3.2 to the following form.
+5.3.5. Corollary. Let E be a locally complete lcHs, G ⊂ E′ determine bound-
+edness, Ω ⊂ Rd open, k ∈ N0 and 0 < γ ≤ 1.
+a) If f∶Ω → E is such that e′ ○f ∈ Ck,γ
+loc (Ω) for all e′ ∈ G, then f ∈ Ck,γ
+loc (Ω,E).
+b) If f∶Ω → E is such that e′ ○f ∈ Ck+1(Ω) for all e′ ∈ G, then f ∈ Ck,1
+loc (Ω,E).
+Proof. Let us start with a). Let f∶Ω → E be such that e′ ○ f ∈ Ck,γ
+loc (Ω) for
+all e′ ∈ G. Let (Ωn)n∈N be an exhaustion of Ω with open, relatively compact sets
+Ωn ⊂ Ω with Lipschitz boundaries ∂Ωn (e.g. choose each Ωn as the interior of a
+finite union of closed axis-parallel cubes, see the proof of [168, Theorem 1.4, p. 7]
+for the construction) that satisfies Ωn ⊂ Ωn+1 for all n ∈ N. Then the restriction of
+e′ ○ f to Ωn is an element of Ck,γ(Ωn) for every e′ ∈ G and n ∈ N. Due to Corollary
+5.3.3 we obtain that f ∈ Ck,γ(Ωn,E) for every n ∈ N. Thus f ∈ Ck,γ
+loc (Ω,E) since
+differentiability is a local property and for each compact K ⊂ Ω there is n ∈ N such
+that K ⊂ Ωn.
+Let us turn to b), i.e. let f∶Ω → E be such that e′ ○ f ∈ Ck+1(Ω) for all e′ ∈ G.
+Since Ω ⊂ Rd is open, for every x ∈ Ω there is εx > 0 such that Bεx(x) ⊂ Ω. For all
+e′ ∈ G, β ∈ Nd
+0 with ∣β∣ = k and w,y ∈ Bεx(x), w ≠ y, it holds that
+∣(∂β)K(e′ ○ f)(w) − (∂β)K(e′ ○ f)(y)∣
+∣w − y∣
+≤ Cd max
+1≤n≤d
+max
+z∈Bεx(x)
+∣(∂β+en)K(e′ ○ f)(z)∣
+by the mean value theorem applied to the real and imaginary part where Cd ∶=
+√
+d
+if K = R, and Cd ∶= 2
+√
+d if K = C.
+Thus e′ ○ f ∈ Ck,1
+loc (Ω) for all e′ ∈ G since
+for each compact set K ⊂ Ω there are m ∈ N and xi ∈ Ω, 1 ≤ i ≤ m, such that
+K ⊂ ⋃m
+i=1 Bεxi (xi). It follows from part a) that f ∈ Ck,1
+loc (Ω,E).
+□
+
+108
+5. APPLICATIONS
+If Ω = R, γ = 1 and G = E′, then part a) of Corollary 5.3.5 is already known
+by [104, 2.3 Corollary, p. 15]. A ‘full’ Ck-weak-strong principle for k < ∞, i.e. the
+conditions of part b) imply f ∈ Ck+1(Ω,E), does not hold in general (see [104, p.
+11–12]) but it holds if we restrict the class of admissible lcHs E.
+5.3.6. Theorem. Let E be a semi-Montel space, G ⊂ E′ determine boundedness,
+Ω ⊂ Rd open and k ∈ N. If f∶Ω → E is such that e′ ○ f ∈ Ck(Ω) for all e′ ∈ G, then
+f ∈ Ck(Ω,E).
+Proof. Let f∶Ω → E be such that e′○f ∈ Ck(Ω) for all e′ ∈ G. Due to Corollary
+5.3.5 b) we already know that f ∈ Ck−1,1
+loc
+(Ω,E) since semi-Montel spaces are quasi-
+complete and thus locally complete. Now, let x ∈ Ω, εx > 0 such that Bεx(x) ⊂ Ω,
+β ∈ Nd
+0 with ∣β∣ = k − 1 and n ∈ N with 1 ≤ n ≤ d. The set
+B ∶= {(∂βf)E(x + hen) − (∂βf)Ef(x)
+h
+∣ h ∈ R, 0 < h ≤ εx}
+is bounded in E because f ∈ Ck−1,1
+loc
+(Ω,E). As E is a semi-Montel space, the closure
+B is compact in E. Let (hm)m∈N be a sequence in R such that 0 < hm ≤ εx for all
+m ∈ N. From the compactness of B we deduce that there is a subnet (hmι)ι∈I of
+(hm)m∈N and yx ∈ B with
+yx = lim
+ι∈I
+(∂βf)E(x + hmιen) − (∂βf)Ef(x)
+hmι
+=∶ lim
+ι∈I yι.
+Further, we note that the limit
+(∂β+en)K(e′ ○ f)(x) =
+lim
+h→0
+h∈R,h≠0
+∂β(e′ ○ f)(x + hen) − ∂β(e′ ○ f)(x)
+h
+(55)
+exists for all e′ ∈ G and that (e′(yι))ι∈I is a subnet of the net of difference quotients
+on the right-hand side of (55) as (∂β)K(e′ ○ f) = e′ ○ (∂β)Ef. Therefore
+(∂β+en)K(e′ ○ f)(x) =
+lim
+h→0
+h∈R,h≠0
+e′((∂β)Ef(x + hen) − (∂β)Ef(x)
+h
+)
+=
+lim
+h→0
+h∈R,0<h≤εx
+e′((∂β)Ef(x + hen) − (∂β)Ef(x)
+h
+) = lim
+ι∈I e′(yι)
+= e′(yx)
+(56)
+for all e′ ∈ G. By Proposition 5.2.56 (i) the topology σ(E,G) and the initial topology
+of E coincide on B. Combining this fact with (56), we deduce that
+(∂β+en)Ef(x) =
+lim
+h→0
+h∈R,h≠0
+(∂β)Ef(x + hen) − (∂β)Ef(x)
+h
+= yx.
+In addition, e′ ○(∂β+en)Ef = (∂β+en)K(e′ ○f) is continuous on Bεx(x) for all e′ ∈ G,
+meaning that the restriction of (∂β+en)Ef on Bεx(x) to (E,σ(E,G)) is continuous,
+and the range (∂β+en)Ef(Bεx(x)) is bounded in E. As before we use that σ(E,G)
+and the initial topology of E coincide on (∂β+en)Ef(Bεx(x)), which implies that
+the restriction of (∂β+en)Ef on Bεx(x) is continuous w.r.t. the initial topology of
+E. Since continuity is a local property and x ∈ Ω is arbitrary, we conclude that
+(∂β+en)Ef is continuous on Ω.
+□
+In the special case that Ω = R, G = E′ and E is a Montel space, i.e. a barrelled
+semi-Montel space, a different proof of the preceding weak-strong principle can be
+found in the proof of [40, Lemma 4, p. 15]. This proof uses the Banach–Steinhaus
+
+5.3. WEAK-STRONG PRINCIPLES FOR DIFFERENTIABILITY OF FINITE ORDER
+109
+theorem and needs the barrelledness of the Montel space E′
+b.
+Our weak-strong
+principle Theorem 5.3.6 does not need the barrelledness of E, hence can be applied
+to E = (C∞
+∂,b(D),β) which is a non-barrelled semi-Montel space by Remark 4.2.23
+and Proposition 5.2.16.
+Besides the ‘full’ Ck-weak-strong principle for k < ∞ and semi-Montel E, part
+b) of Corollary 5.3.5 also suggests an ‘almost’ Ck-weak-strong principle in terms of
+[66, 3.1.6 Rademacher’s theorem, p. 216], which we prepare next.
+5.3.7. Definition (generalised Gelfand space). We say that an lcHs E is a
+generalised Gelfand space if every Lipschitz continuous map f∶[0,1] → E is differ-
+entiable almost everywhere w.r.t to the one-dimensional Lebesgue measure.
+If E is a real Fréchet space (K = R), then this definition coincides with the
+definition of a Fréchet–Gelfand space given in [125, Definition 2.2, p. 17]. In par-
+ticular, every real nuclear Fréchet lattice (see [78, Theorem 6, Corollary, p. 375,
+378]) and more general every real Fréchet–Montel space is a generalised Gelfand
+space by [125, Theorem 2.9, p. 18]. If E is a Banach space, then this definition
+coincides with the definition of a Gelfand space given in [51, Definition 4.3.1, p.
+106–107] by [9, Proposition 1.2.4, p. 18]. A Banach space is a Gelfand space if and
+only if it has the Radon–Nikodým property by [51, Theorem 4.3.2, p. 107]. Thus
+separable duals of Banach spaces, reflexive Banach spaces and ℓ1(Γ) for any set
+Γ are generalised Gelfand spaces by [51, Theorem 3.3.1 (Dunford-Pettis), p. 79],
+[51, Corollary 3.3.4 (Phillips), p. 82] and [51, Corollary 3.3.8, p. 83]. The Banach
+spaces c0, ℓ∞, C([0,1]), L1([0,1]) and L∞([0,1]) do not have the Radon–Nikodým
+property and hence are not generalised Gelfand spaces by [9, Proposition 1.2.9, p.
+20], [9, Example 1.2.8, p. 20] and [9, Proposition 1.2.10, p. 21].
+5.3.8. Corollary. Let E be a locally complete generalised Gelfand space, G ⊂
+E′ determine boundedness, Ω ⊂ R open and k ∈ N. If f∶Ω → E is such that e′ ○ f ∈
+Ck(Ω) for all e′ ∈ G, then f ∈ Ck−1,1
+loc
+(Ω,E) and the derivative (∂k)Ef(x) exists for
+Lebesgue almost all x ∈ Ω.
+Proof. The first part follows from Corollary 5.3.5 b). Now, let [a,b] ⊂ Ω be
+a bounded interval. We set F∶[0,1] → E, F(x) ∶= (∂k−1)Ef(a + x(b − a)). Then F
+is Lipschitz continuous as f ∈ Ck−1,1
+loc
+(Ω,E). This yields that F is differentiable on
+[0,1] almost everywhere because E is a generalised Gelfand space, implying that
+(∂k−1)Ef is differentiable on [a,b] almost everywhere. Since the open set Ω ⊂ R
+can be written as a countable union of disjoint open intervals In, n ∈ N, and each
+In is a countable union of closed bounded intervals [am,bm], m ∈ N, our statement
+follows from the fact that the countable union of null sets is a null set.
+□
+To the best of our knowledge there are still some open problems for continuously
+partially differentiable functions of finite order.
+5.3.9. Question.
+(i) Are there other spaces than semi-Montel spaces E
+for which the ‘full’ Ck-weak-strong principle Theorem 5.3.6 with k < ∞
+is true? For instance, if k = 0, then it is still true if E is a generalised
+Schwartz space by [17, 2.10 Lemma, p. 140]. Does this hold for 0 < k < ∞
+as well?
+(ii) Does the ‘almost’ Ck-weak-strong principle Corollary 5.3.8 also hold for
+d > 1?
+(iii) For every ε > 0 does there exist a function g ∈ Ck(R,E) such that λ({x ∈
+Ω ∣ f(x) ≠ g(x)}) < ε in Corollary 5.3.8 where λ is the one-dimensional
+Lebesgue measure. In the case that E = Rn this is true by [66, Theorem
+3.1.15, p. 227].
+
+110
+5. APPLICATIONS
+(iv) Is there a ‘Radon–Nikodým type’ characterisation of generalised Gelfand
+spaces as in the Banach case?
+5.4. Vector-valued Blaschke theorems
+In this section we prove several convergence theorems for Banach-valued func-
+tions in the spirit of Blaschke’s convergence theorem [38, Theorem 7.4, p. 219]
+as it is done in [7, Theorem 2.4, p. 786] and [7, Corollary 2.5, p. 786–787] for
+bounded holomorphic functions and more general in [70, Corollary 4.2, p. 695] for
+bounded functions in the kernel of a hypoelliptic linear partial differential operator.
+Blaschke’s convergence theorem says that if (zn)n∈N ⊂ D is a sequence of distinct
+elements with ∑n∈N(1 − ∣zn∣) = ∞ and if (fk)k∈N is a bounded sequence in H∞(D)
+such that (fk(zn))k converges in C for each n ∈ N, then there is f ∈ H∞(D) such
+that (fk)k converges uniformly to f on the compact subsets of D, i.e. w.r.t. to τc.
+5.4.1. Proposition ([70, Proposition 4.1, p. 695]). Let (E,∥ ⋅ ∥) be a Banach
+space, Z a Banach space whose closed unit ball BZ is a compact subset of an lcHs
+Y and let (Aι)ι∈I be a net in Y εE such that
+sup
+ι∈I
+{∥Aι(y)∥ ∣ y ∈ B○Y ′
+Z
+} < ∞.
+Assume further that there exists a σ(Y ′,Z)-dense subspace X ⊂ Y ′ such that
+limι Aι(x) exists for each x ∈ X. Then there is A ∈ Y εE with A(B○Y ′
+Z
+) bounded
+and limι Aι = A uniformly on the equicontinuous subsets of Y ′, i.e. for all equicon-
+tinuous B ⊂ Y ′ and ε > 0 there exists ς ∈ I such that
+sup
+y∈B
+∥Aι(y) − A(y)∥ < ε
+for each ι ≥ ς.
+Next, we generalise [70, Corollary 4.2, p. 695].
+5.4.2. Corollary. Let (E,∥ ⋅ ∥) be a Banach space and F(Ω) and F(Ω,E) be
+ε-into-compatible. Let (T E,T K) be a generator for (Fν,E) and a strong, consistent
+family for (F,E), Fν(Ω) a Banach space whose closed unit ball BFν(Ω) is a compact
+subset of F(Ω) and U a set of uniqueness for (T K,Fν).
+If (fι)ι∈I ⊂ Fεν(Ω,E) is a bounded net in Fν(Ω,E) such that limι T E(fι)(x)
+exists for all x ∈ U, then there is f ∈ Fεν(Ω,E) such that (fι)ι∈I converges to f in
+F(Ω,E).
+Proof. We set X ∶= span{T K
+x ∣ x ∈ U}, Y ∶= F(Ω) and Z ∶= Fν(Ω). As in
+the proof of Theorem 5.2.29 we observe that X is σ(Y ′,Z)-dense in Y ′.
+From
+(fι)ι∈I ⊂ Fεν(Ω,E) follows that there are Aι ∈ F(Ω)εE with S(Aι) = fι for all ι ∈ I.
+Since (fι)ι∈I is a bounded net in Fν(Ω,E), we note that
+sup
+ι∈I
+sup
+x∈ω ∥Aι(T K
+x (⋅)ν(x))∥ = sup
+ι∈I
+sup
+x∈ω ∥T ES(Aι)(x)∥ν(x) = sup
+ι∈I
+sup
+x∈ω ∥T Efι(x)∥ν(x)
+= sup
+ι∈I
+∣fι∣Fν(Ω,E) < ∞
+by consistency. Further, limι S(Aι)(T K
+x ) = limι T E(fι)(x) exists for each x ∈ U,
+implying the existence of limι S(Aι)(x) for each x ∈ X by linearity.
+We apply
+Proposition 5.4.1 and obtain f ∶= S(A) ∈ Fεν(Ω,E) such that (Aι)ι∈I converges
+to A in F(Ω)εE. From F(Ω) and F(Ω,E) being ε-into-compatible it follows that
+(fι)ι∈I converges to f in F(Ω,E).
+□
+
+5.4. VECTOR-VALUED BLASCHKE THEOREMS
+111
+First, we apply the preceding corollary to the space C[γ]
+z (Ω,E) of γ-Hölder
+continuous functions on Ω that vanish at a fixed point z ∈ Ω from Example 4.2.9
+a). We recall that for a metric space (Ω,d), z ∈ Ω, an lcHs E and 0 < γ ≤ 1 we have
+C[γ]
+z (Ω,E) = {f ∈ EΩ ∣ f(z) = 0 and ∀ α ∈ A ∶ ∣f∣C0,γ(Ω),α < ∞}.
+Further, we set ω ∶= Ω2 ∖ {(x,x) ∣ x ∈ Ω}, F(Ω,E) ∶= {f ∈ C(Ω,E) ∣ f(z) = 0} and
+T E∶F(Ω,E) → Eω, T E(f)(x,y) ∶= f(x) − f(y), and
+ν∶ω → [0,∞), ν(x,y) ∶=
+1
+d(x,y)γ .
+Then we have for every α ∈ A that
+∣f∣C0,γ(Ω),α = sup
+x∈ω pα(T E(f)(x))ν(x),
+f ∈ C[γ]
+z (Ω,E),
+and observe that Fν(Ω,E) = C[γ]
+z (Ω,E) with generator (T E,T K).
+5.4.3. Corollary. Let E be a Banach space, (Ω,d) a metric space, z ∈ Ω and
+0 < γ ≤ 1. If (fι)ι∈I is a bounded net in C[γ]
+z (Ω,E) such that limι fι(x) exists for all
+x in a dense subset U ⊂ Ω, then there is f ∈ C[γ]
+z (Ω,E) such that (fι)ι∈I converges
+to f in C(Ω,E) uniformly on compact subsets of Ω.
+Proof. We choose F(Ω) ∶= {f ∈ C(Ω) ∣ f(z) = 0} and F(Ω,E) ∶= {f ∈
+C(Ω,E) ∣ f(z) = 0}. Then we have Fν(Ω) = C[γ]
+z (Ω) and Fν(Ω,E) = C[γ]
+z (Ω,E)
+with the weight ν and generator (T E,T K) for (Fν,E) described above. Due to [17,
+3.1 Bemerkung, p. 141] the spaces F(Ω) and F(Ω,E), equipped with the topology
+τc of compact convergence, are ε-compatible. Obviously, (T E,T K) is a strong, con-
+sistent family for (F,E). In addition, Fν(Ω) = C[γ]
+z (Ω) is a Banach space by [179,
+Proposition 1.6.2, p. 20]. For all f from the closed unit ball BFν(Ω) of Fν(Ω) we
+have
+∣f(x) − f(y)∣ ≤ d(x,y)γ,
+x,y ∈ Ω,
+and
+∣f(x)∣ = ∣f(x) − f(z)∣ ≤ d(x,z)γ,
+x ∈ Ω.
+It follows that BFν(Ω) is (uniformly) equicontinuous and {f(x) ∣ f ∈ BFν(Ω)} is
+bounded in K for all x ∈ Ω. Ascoli’s theorem (see e.g. [133, Theorem 47.1, p. 290])
+implies the compactness of BFν(Ω) in F(Ω) (see also [118, 3.7 Theorem (a), p. 10]).
+Furthermore, the ε-compatibility of F(Ω) and F(Ω,E) in combination with the
+consistency of (T E,T K) for (F,E) gives Fεν(Ω,E) = Fν(Ω,E) as linear spaces by
+Proposition 5.2.25 c). We note that limι fι(x) = limι T E(fι)(x,z) for all x in U,
+proving our claim by Corollary 5.4.2.
+□
+The space C[γ]
+z (Ω) is named Lip0(Ωγ) in [179] (see [179, Definition 1.6.1 (b), p.
+19] and [179, Definition 1.1.2, p. 2]). Corollary 5.4.3 generalises [179, Proposition
+2.1.7, p. 38] (in combination with [179, Proposition 1.2.4, p. 5]) where Ω is compact,
+U = Ω and E = K.
+5.4.4. Corollary. Let E be a Banach space, Ω ⊂ Rd open and bounded, k ∈ N0
+and 0 < γ ≤ 1. In the case k ≥ 1, assume additionally that Ω has Lipschitz boundary.
+If (fι)ι∈I is a bounded net in Ck,γ(Ω,E) such that
+(i) limι fι(x) exists for all x in a dense subset U ⊂ Ω, or if
+(ii) limι(∂en)Efι(x) exists for all 1 ≤ n ≤ d and x in a dense subset U ⊂ Ω, Ω
+is connected and there is x0 ∈ Ω such that limι fι(x0) exists and k ≥ 1,
+then there is f ∈ Ck,γ(Ω,E) such that (fι)ι∈I converges to f in Ck(Ω,E).
+
+112
+5. APPLICATIONS
+Proof. As in Corollary 5.3.3 we take F(Ω) ∶= Ck(Ω) and F(Ω,E) ∶= Ck(Ω,E)
+as well as Fν(Ω) ∶= Ck,γ(Ω) and Fν(Ω,E) ∶= Ck,γ(Ω,E) with the weight ν and
+generator (T E,T K) for (Fν,E) described above of Corollary 5.3.3. By the proof
+of Corollary 5.3.3 all conditions of Corollary 5.4.2 are satisfied, which implies our
+statement.
+□
+We recall that CWk(Ω,E) is the space Ck(Ω,E) equipped with its usual topol-
+ogy for an open set Ω ⊂ Rd, k ∈ N∞ ∪ {0} and an lcHs E (see Example 3.1.9 b) for
+k ∈ N∞ and the definition above Proposition 3.1.11 for k = 0).
+5.4.5. Corollary. Let E be a Banach space, Ω ⊂ Rd open, k ∈ N0 and 0 < γ ≤ 1.
+If (fι)ι∈I is a bounded net in Ck,γ
+loc (Ω,E) such that
+(i) limι fι(x) exists for all x in a dense subset U ⊂ Ω, or if
+(ii) limι(∂en)Efι(x) exists for all 1 ≤ n ≤ d and x in a dense subset U ⊂ Ω, Ω
+is connected and there is x0 ∈ Ω such that limι fι(x0) exists and k ≥ 1,
+then there is f ∈ Ck,γ
+loc (Ω,E) such that (fι)ι∈I converges to f in CWk(Ω,E).
+Proof. Let (Ωn)n∈N be an exhaustion of Ω with open, relatively compact sets
+Ωn ⊂ Ω such that Ωn has Lipschitz boundary, Ωn ⊂ Ωn+1 for all n ∈ N and, in
+addition, x0 ∈ Ω1 and Ωn is connected for each n ∈ N in case (ii) (see the proof of
+Corollary 5.3.5). The restriction of (fι)ι∈I to Ωn is a bounded net in Ck,γ(Ωn,E)
+for each n ∈ N. By Corollary 5.4.4 there is Fn ∈ Ck,γ(Ωn,E) for each n ∈ N such
+that the restriction of (fι)ι∈I to Ωn converges to Fn in Ck(Ωn,E) since U ∩ Ωn is
+dense in Ωn due to Ωn being open and x0 being an element of the connected set
+Ωn in case (ii). The limits Fn+1 and Fn coincide on Ωn for each n ∈ N. Thus the
+definition f ∶= Fn on Ωn for each n ∈ N gives a well-defined function f ∈ Ck,γ
+loc (Ω,E),
+which is a limit of (fι)ι∈I in CWk(Ω,E).
+□
+5.4.6. Corollary. Let E be a Banach space, Ω ⊂ Rd open and k ∈ N0. If
+(fι)ι∈I is a bounded net in Ck+1(Ω,E) such that
+(i) limι fι(x) exists for all x in a dense subset U ⊂ Ω, or if
+(ii) limι(∂en)Efι(x) exists for all 1 ≤ n ≤ d and x in a dense subset U ⊂ Ω, Ω
+is connected and there is x0 ∈ Ω such that limι fι(x0) exists,
+then there is f ∈ Ck,1
+loc (Ω,E) such that (fι)ι∈I converges to f in CWk(Ω,E).
+Proof. By Corollary 5.3.5 b) (fι)ι∈I is a bounded net in Ck,1
+loc (Ω,E). Hence
+our statement is a consequence of Corollary 5.4.5.
+□
+The preceding result directly implies a C∞-smooth version.
+5.4.7. Corollary. Let E be a Banach space and Ω ⊂ Rd open. If (fι)ι∈I is a
+bounded net in C∞(Ω,E) such that
+(i) limι fι(x) exists for all x in a dense subset U ⊂ Ω, or if
+(ii) limι(∂en)Efι(x) exists for all 1 ≤ n ≤ d and x in a dense subset U ⊂ Ω, Ω
+is connected and there is x0 ∈ Ω such that limι fι(x0) exists,
+then there is f ∈ C∞(Ω,E) such that (fι)ι∈I converges to f in CW∞(Ω,E).
+Now, we turn to weighted kernels of hypoelliptic linear partial differential op-
+erators.
+5.4.8. Corollary. Let E be a Banach space, Ω ⊂ Rd open, P(∂)K a hypoelliptic
+linear partial differential operator, ν∶Ω → (0,∞) continuous and U ⊂ Ω a set of
+uniqueness for (idKΩ,CνP (∂)). If (fι)ι∈I is a bounded net in (CνP (∂)(Ω,E),∣ ⋅ ∣ν)
+such that limι fι(x) exists for all x ∈ U, then there is f ∈ CνP (∂)(Ω,E) such that
+(fι)ι∈I converges to f in (C∞
+P (∂)(Ω,E),τc).
+
+5.4. VECTOR-VALUED BLASCHKE THEOREMS
+113
+Proof. Our statement follows from Corollary 5.4.2 since by the proof of Corol-
+lary 5.2.30 all conditions needed are fulfilled.
+□
+For ν = 1 on Ω the preceding corollary is included in [70, Corollary 4.2, p.
+695] but then an even better result is available, whose proof we prepare next.
+We recall the definition of the space (C∞
+P (∂),b(Ω,E),β) with the strict topology β
+from Proposition 4.2.24. For an open set Ω ⊂ Rd, an lcHs E and a linear partial
+differential operator P(∂)E which is hypoelliptic if E = K the space of bounded
+zero solutions is
+C∞
+P (∂),b(Ω,E) = {f ∈ C∞
+P (∂)(Ω,E) ∣ ∀ α ∈ A ∶ ∥f∥∞,α = sup
+x∈Ω
+pα(f(x)) < ∞}.
+We equip this space with strict topology β induced by the seminorms
+∣f∣̃ν,α ∶= sup
+x∈Ω
+pα(f(x))∣̃ν(x)∣,
+f ∈ C∞
+P (∂),b(Ω,E),
+for ̃ν ∈ C0(Ω). Now, we phrase for C∞
+P (∂),b(Ω,E) = CνP (∂)(Ω,E) with ν = 1 on Ω
+the improved version of Corollary 5.4.8.
+5.4.9. Corollary. Let E be a Banach space, Ω ⊂ Rd open, P(∂)K a hypoelliptic
+linear partial differential operator and U ⊂ Ω a set of uniqueness for (idKΩ,C∞
+P (∂),b).
+If (fι)ι∈I is a bounded net in (C∞
+P (∂),b(Ω,E),∥ ⋅ ∥∞) such that limι fι(x) exists
+for all x ∈ U, then there is f ∈ C∞
+P (∂),b(Ω,E) such that (fι)ι∈I converges to f in
+(C∞
+P (∂),b(Ω,E),β).
+Proof. We take F(Ω) ∶= (C∞
+P (∂),b(Ω),β) and F(Ω,E) ∶= (C∞
+P (∂),b(Ω,E),β) as
+well as Fν(Ω) ∶= (C∞
+P (∂),b(Ω),∥ ⋅ ∥∞) and Fν(Ω,E) ∶= (C∞
+P (∂),b(Ω,E),∥ ⋅ ∥∞) with
+the weight ν(x) ∶= 1, x ∈ Ω, and generator (idEΩ,idΩK) for (Fν,E). The generator
+is strong and consistent for (F,E) and F(Ω) and F(Ω,E) are ε-compatible by
+Proposition 4.2.24. The space Fν(Ω) is a Banach space as a closed subspace of
+the Banach space (Cb(Ω),∥ ⋅ ∥∞). Its closed unit ball BFν(Ω) is τc-compact because
+(C∞
+P (∂)(Ω),τc) is a Fréchet–Schwartz space, in particular, a Montel space. Thus
+BFν(Ω) is ∥ ⋅ ∥∞-bounded and τc-compact, which implies that it is also β-compact
+by [45, Proposition 1 (viii), p. 586] and [45, Proposition 3, p. 590]. In addition,
+the ε-compatibility of F(Ω) and F(Ω,E) in combination with the consistency of
+(idEΩ,idKΩ) for (F,E) gives Fεν(Ω,E) = Fν(Ω,E) as linear spaces by Proposition
+5.2.25 c), verifying our statement by Corollary 5.4.2.
+□
+A direct consequence of Corollary 5.4.9 is the following remark.
+5.4.10. Remark. Let E be a Banach space, Ω ⊂ Rd open, P(∂)K a hypoel-
+liptic linear partial differential operator and (fι)ι∈I a bounded net in the space
+(C∞
+P (∂),b(Ω,E),∥ ⋅ ∥∞). Then the following statements are equivalent:
+(i) (fι) converges pointwise,
+(ii) (fι) converges uniformly on compact subsets of Ω,
+(iii) (fι) is β-convergent.
+In the case of complex-valued bounded holomorphic functions of one variable,
+i.e. E = C, Ω ⊂ C is open and P(∂) = ∂ is the Cauchy–Riemann operator, conver-
+gence w.r.t. β is known as bounded convergence (see [147, p. 13–14, 16]) and the
+preceding remark is included in [148, 3.7 Theorem, p. 246] for connected sets Ω.
+A similar improvement of Corollary 5.4.3 for the space C[γ]
+z (Ω,E) of γ-Hölder
+continuous functions on a metric space (Ω,d) that vanish at a given point z ∈ Ω is
+
+114
+5. APPLICATIONS
+possible, using the strict topology β on C[γ]
+z (Ω) given by the seminorms
+∣f∣ν ∶= sup
+x,y∈Ω
+x≠y
+∣f(x) − f(y)∣
+∣x − y∣γ
+∣ν(x,y)∣,
+f ∈ C[γ]
+z (Ω),
+for ν ∈ C0(ω) with ω = Ω2 ∖ {(x,x) ∣ x ∈ Ω}. If Ω is compact and E a Banach
+space, this follows as in Corollary 5.4.9 from the observation that β is the mixed
+topology γ(∣ ⋅ ∣C0,γ(Ω),τc) by [90, Theorem 3.3, p. 645], that a set is β-compact if
+and only if it is ∣⋅∣C0,γ(Ω)-bounded and τc-compact by [90, Theorem 2.1 (6), p. 642],
+the ε-compatibility (C[γ]
+z (Ω),β)εE ≅ (C[γ]
+z (Ω,E),γτγ) by [90, Theorem 4.4, p. 648]
+where the topology γτγ is described in [90, Definition 4.1, p. 647] and coincides
+with β if E = K by [90, Proposition 4.3 (i), p. 647].
+Let us turn to Bloch type spaces. The result corresponding to Corollary 5.4.8
+for Bloch type spaces reads as follows.
+5.4.11. Corollary. Let E be a Banach space, ν∶D → (0,∞) continuous and
+U∗ ⊂ D have an accumulation point in D. If (fι)ι∈I is a bounded net in Bν(D,E)
+such that limι fι(0) and limι(∂1
+C)Efι(z) exist for all z ∈ U∗, then there is f ∈
+Bν(D,E) such that (fι)ι∈I converges to f in (O(D,E),τc).
+Proof. Due to the proof of Corollary 5.2.33 all conditions needed to apply
+Corollary 5.4.2 are fulfilled, which proves our statement.
+□
+5.5. Wolff type results
+The following theorem gives us a Wolff type description of the dual of F(Ω) and
+generalises [70, Theorem 3.3, p. 693] and [70, Corollary 3.4, p. 694] whose proofs
+only need a bit of adaptation. Wolff’s theorem [183, p. 1327] (cf. [81, Theorem
+(Wolff), p. 402]) phrased in a functional analytic way (see [70, p. 240]) says: if
+Ω ⊂ C is a domain (i.e. open and connected), then for each µ ∈ O(Ω)′ there are
+a sequence (zn)n∈N which is relatively compact in Ω and a sequence (an)n∈N in ℓ1
+such that µ = ∑∞
+n=1 anδzn.
+5.5.1. Theorem. Let F(Ω) and F(Ω,E) be ε-into-compatible, (T E,T K) be a
+generator for (Fν,E) and a strong, consistent family for (F,E) for every Banach
+space E. Let F(Ω) be a nuclear Fréchet space and Fν(Ω) a Banach space whose
+closed unit ball BFν(Ω) is a compact subset of F(Ω) and (xn)n∈N fixes the topology
+in Fν(Ω).
+a) Then there is 0 < λ ∈ ℓ1, i.e. λ ∈ ℓ1 and λn > 0 for all n ∈ N, such that for
+every bounded B ⊂ F(Ω)′
+b there is C ≥ 1 with
+{µ∣Fν(Ω) ∣ µ ∈ B} ⊂ {
+∞
+∑
+n=1
+anν(xn)T K
+xn ∈ Fν(Ω)′ ∣ a ∈ ℓ1, ∀ n ∈ N ∶ ∣an∣ ≤ Cλn}.
+b) Let (∥ ⋅ ∥k)k∈N denote the system of seminorms generating the topology of
+F(Ω). Then there is a decreasing zero sequence (εn)n∈N such that for all
+k ∈ N there is C ≥ 1 with
+∥f∥k ≤ C sup
+n∈N
+∣T K(f)(xn)∣ν(xn)εn,
+f ∈ Fν(Ω).
+Proof. We start with part a).
+Let B1 ∶= {T K
+xn(⋅)ν(xn) ∣ n ∈ N} ⊂ F(Ω)′,
+X ∶= spanB1, Y ∶= F(Ω), Z ∶= Fν(Ω) and E1 ∶= {∑∞
+n=1 anν(xn)T K
+xn ∣ a ∈ ℓ1}. From
+∣j1(a)(f)∣ ∶= ∣
+∞
+∑
+n=1
+anν(xn)T K
+xn(f)∣ ≤ sup
+n∈N
+∣T K(f)(xn)∣ν(xn)∥a∥ℓ1 ≤ ∣f∣Fν(Ω)∥a∥ℓ1
+for all f ∈ Fν(Ω) and a ∈ ℓ1 it follows that E1 is a linear subspace of Fν(Ω)′
+and the continuity of the map j1∶ℓ1 → Fν(Ω)′ where Fν(Ω)′ is equipped with the
+
+5.5. WOLFF TYPE RESULTS
+115
+operator norm. In addition, we deduce that the linear map j∶ℓ1/kerj1 → Fν(Ω)′,
+j([a]) ∶= j1(a), where [a] denotes the equivalence class of a ∈ ℓ1 in the quotient
+space ℓ1/kerj1, is continuous w.r.t. the quotient norm since
+∥j([a])∥Fν(Ω)′ ≤
+inf
+b∈ℓ1,[b]=[a]∥b∥ℓ1 = ∥[a]∥ℓ1/ ker j1.
+By setting E ∶= j(ℓ1/kerj1) and ∥j([a])∥E ∶= ∥[a]∥ℓ1/ ker j1, a ∈ ℓ1, and observing
+that ℓ1/kerj1 is a Banach space, we obtain that E is also a Banach space, which is
+continuously embedded in Fν(Ω)′.
+We denote by A∶X → E the restriction to Z = Fν(Ω) determined by
+A(T K
+xn(⋅)ν(xn)) ∶= T K
+xn(⋅)∣Fν(Ω)ν(xn) = j([en])
+where en is the n-th unit sequence in ℓ1. We consider Fν(Ω) as a subspace of E′
+via
+f(j([a])) ∶= j([a])(f) =
+∞
+∑
+n=1
+anν(xn)T K(f)(xn),
+a ∈ ℓ1,
+for f ∈ Fν(Ω). The space G ∶= Fν(Ω) clearly separates the points of E, thus is
+σ(E′,E)-dense and
+(f ○ A)(T K
+xn(⋅)ν(xn)) = A(T K
+xn(⋅)ν(xn))(f) = j([en])(f) = f(j([en]))
+for all n ∈ N. Hence we may consider f ○ A by identification with f as an element
+of Z = Fν(Ω) for all f ∈ G = Fν(Ω). It follows from Proposition 5.2.68 that there
+is a unique extension ̂A ∈ F(Ω)εE of A such that S(̂A) ∈ Fεν(Ω,E).
+For each e′ ∈ E′ there are C0,C1 > 0 and an absolutely convex compact set
+K ⊂ F(Ω) such that
+∣(e′ ○ ̂A)(µ)∣ ≤ C0∥̂A(µ)∥E ≤ C0C1 sup
+f∈K
+∣µ(f)∣
+for all µ ∈ F(Ω)′, implying e′ ○ ̂A ∈ (F(Ω)′
+b)′. Due to the reflexivity of the nuclear
+Fréchet space F(Ω) we obtain e′ ○ ̂A ∈ F(Ω) for each e′ ∈ E′. Further, for each
+e′ ∈ E′ we have
+∥e′ ○ ̂A∥Fν(Ω) = sup
+x∈ω ∣T K(e′ ○ ̂A)(x)∣ν(x) = sup
+x∈ω ∣(e′ ○ ̂A)(T K
+x (⋅)ν(x))∣
+≤ C0 sup
+x∈ω ∥̂A(T K
+x (⋅)ν(x))∥E < ∞
+since ̂A(B○F (Ω)′
+Fν(Ω) ) is bounded in E. This yields e′ ○ ̂A ∈ Fν(Ω) for each e′ ∈ E′. In
+particular, we get that ̂A is σ(F(Ω)′,Fν(Ω))-σ(E,E′) continuous. The restriction
+r∶F(Ω)′ → Fν(Ω)′, r(µ) ∶= µ∣Fν(Ω), is σ(F(Ω)′,Fν(Ω))-σ(Fν(Ω)′,Fν(Ω)) contin-
+uous and coincides with ̂A on the σ(F(Ω)′,Fν(Ω))-dense subspace X = spanB1 ⊂
+F(Ω)′. Therefore ̂A(µ) = r(µ) = µ∣Fν(Ω) for all µ ∈ F(Ω)′.
+Let B be an absolutely convex, closed and bounded subset of F(Ω)′
+b.
+We
+endow W ∶= spanB with the Minkowski functional of B. Due to the nuclearity
+of F(Ω), there are an absolutely convex, closed and bounded subset V ⊂ F(Ω)′
+b,
+(w′
+k)k∈N ⊂ BW ′, (µk)k∈N ⊂ V and 0 ≤ γ ∈ ℓ1, i.e. γ ∈ ℓ1 and γn ≥ 0 for all n ∈ N, such
+that
+µ =
+∞
+∑
+k=1
+γkw′
+k(µ)µk,
+µ ∈ B,
+by [24, 2.9.1 Theorem, p. 134, 2.9.2 Definition, p. 135]. The boundedness of ̂A(V )
+in E and the definition of E give us a bounded sequence ([β(k)])k∈N ⊂ E with
+µk∣Fν(Ω) = ̂A(µk) =
+∞
+∑
+n=1
+β(k)
+n ν(xn)T K
+xn
+
+116
+5. APPLICATIONS
+for all k ∈ N. The sequence (β(k))k∈N ⊂ ℓ1 is also bounded by [131, Remark 5.11, p.
+36] and we set ρn ∶= ∑∞
+k=1 γk∣β(k)
+n ∣ for n ∈ N. With ρ ∶= (ρn)n∈N we have
+∥ρ∥ℓ1 =
+∞
+∑
+n=1
+∞
+∑
+k=1
+γk∣β(k)
+n ∣ ≤
+∞
+∑
+n=1
+sup
+l∈N
+∣β(l)
+n ∣
+∞
+∑
+k=1
+γk = sup
+l∈N
+∥β(l)∥ℓ1∥γ∥ℓ1 < ∞,
+which means that ρ ∈ ℓ1. For every µ ∈ B we set an ∶= ∑∞
+k=1 γkw′
+k(µ)β(k)
+n , n ∈ N,
+and conclude that a ∈ ℓ1 with ∣an∣ ≤ ρn for all n ∈ N and
+µ∣Fν(Ω) =
+∞
+∑
+n=1
+anν(xn)T K
+xn.
+(57)
+The strong dual F(Ω)′
+b of the Fréchet–Schwartz space F(Ω) is a DFS-space and
+thus there is a fundamental sequence of bounded (closed, absolutely convex) sets
+(Bl)l∈N in F(Ω)′
+b by [131, Proposition 25.19, p. 303]. Due to our preceding results
+there is ρ(l) ∈ ℓ1 with (57) for each l ∈ N. Finally, part a) follows from choosing
+0 < λ ∈ ℓ1 such that each ρ(l) is componentwise smaller than a multiple of λ, i.e.
+we choose λ in a way that for each l ∈ N there is Cl ≥ 1 with ρ(l)
+n
+≤ Clλn for all
+n ∈ N (w.l.o.g. we may assume (the worst case) that limn→∞ ρ(l+1)
+n
+/ρ(l)
+n = ∞ for each
+l ∈ N. Then the construction of a suitable 0 < λ ∈ ℓ1 is given in [97, Chap. IX, §41,
+7., p. 301–302]: set c(l)
+n ∶= ρ(l)
+n
+for all l,n ∈ N and define λn ∶= cn +
+1
+n2 for all n ∈ N
+with the (cn) ∈ ℓ1 constructed there. Then set C1 ∶= 1 and Cl ∶= (max{c(l)
+n ∣ 1 ≤ n ≤
+nl−1}/min{λn ∣ 1 ≤ n ≤ nl−1}) + 1, l ≥ 2, for the sequence of indices (nl)l∈N from the
+construction of (cn).).
+Let us turn to part b). We choose λ ∈ ℓ1 from part a) and a decreasing zero
+sequence (εn)n∈N such that ( λn
+εn )n∈N still belongs to ℓ1 (e.g. take εn ∶= (∑∞
+k=n λk)1/2
+for n ∈ N by [97, Chap. IX, §39, Theorem of Dini, p. 293]). For k ∈ N we set
+̃Bk ∶= {f ∈ F(Ω) ∣ ∥f∥k ≤ 1}
+and note that the polar ̃B○
+k is bounded in F(Ω)′
+b. Due to part a) there exists C ≥ 1
+such that
+̂A( ̃B○
+k) ⊂ {
+∞
+∑
+n=1
+anν(xn)T K
+xn ∈ Fν(Ω)′ ∣ a ∈ ℓ1, ∀ n ∈ N ∶ ∣an∣ ≤ Cλn}.
+By [131, Proposition 22.14, p. 256] the formula
+∥f∥k = sup
+y′∈ ̃
+B○
+k
+∣y′(f)∣,
+f ∈ F(Ω),
+is valid and hence
+∥f∥k = sup
+y′∈ ̃
+B○
+k
+∣r(y′)(f)∣ = sup
+y′∈ ̃
+B○
+k
+∣̂A(y′)(f)∣ ≤ C sup
+a∈ℓ1
+∣an∣≤λn
+∣
+∞
+∑
+n=1
+anν(xn)T K(f)(xn)∣
+≤ C∥(λn
+εn
+)
+n∥
+ℓ1 sup
+n∈N
+∣T K(f)(xn)∣ν(xn)εn
+for all f ∈ Fν(Ω).
+□
+5.5.2. Remark. The proof of Theorem 5.5.1 shows it is not needed that the
+assumption that F(Ω) and F(Ω,E) are ε-into-compatible, (T E,T K) is a generator
+for (Fν,E) and a strong, consistent family for (F,E) is fulfilled for every Banach
+space E. It is sufficient that it is fulfilled for the Banach space E ∶= j(ℓ1/kerj1).
+We recall from (53) that for a positive sequence ν ∶= (νn)n∈N and an lcHs E we
+have
+ℓν(N,E) = {x = (xn)n∈N ∈ EN ∣ ∀ α ∈ A ∶ ∥x∥α = sup
+n∈N
+pα(xn)νn < ∞}.
+
+5.6. SERIES REPRESENTATION OF VECTOR-VALUED FUNCTIONS
+117
+Further, we equip the space EN of all sequences in E from Example 4.2.1 with the
+topology of pointwise convergence, i.e. the topology generated by the seminorms
+∣x∣k,α ∶= sup
+1≤n≤k
+pα(xn),
+x = (xn)n∈N ∈ EN,
+for k ∈ N and α ∈ A.
+5.5.3. Corollary. Let ν ∶= (νn)n∈N be a positive sequence.
+a) Then there is 0 < λ ∈ ℓ1 such that for every bounded B ⊂ (KN)′
+b there is
+C ≥ 1 with
+{µ∣ℓν(N) ∣ µ ∈ B} ⊂ {
+∞
+∑
+n=1
+anνnδn ∈ ℓν(N)′ ∣ a ∈ ℓ1, ∀ n ∈ N ∶ ∣an∣ ≤ Cλn}.
+b) Then there is a decreasing zero sequence (εn)n∈N such that for all k ∈ N
+there is C ≥ 1 with
+sup
+1≤n≤k
+∣xn∣ ≤ C sup
+n∈N
+∣xn∣νnεn,
+x = (xn)n∈N ∈ ℓν(N).
+Proof. We take F(N) ∶= KN and F(N,E) ∶= EN as well as Fν(N) ∶= ℓν(N) and
+Fν(N,E) ∶= ℓν(N,E) where (T E,T K) ∶= (idEN,idKN) is the generator for (Fν,E).
+We remark that F(N) and F(N,E) are ε-compatible and (T E,T K) is a strong,
+consistent family for (F,E) by Example 4.2.1 for every Banach space E. Moreover,
+Fν(N) = ℓν(N) is a Banach space by [131, Lemma 27.1, p. 326] since ℓν(N) = λ∞(A)
+with the Köthe matrix A ∶= (an,j)n,j∈N given by an,j ∶= νn for all n,j ∈ N.
+In
+addition, we have for every k ∈ N
+sup
+1≤n≤k
+∣xn∣ ≤ sup
+1≤n≤k
+ν−1
+n ∣x∣ν ≤ sup
+1≤n≤k
+ν−1
+n ,
+x = (xn)n∈N ∈ BFν(N),
+which means that BFν(N) is bounded in F(N). The space F(N) = KN is a nuclear
+Fréchet space and BFν(N) is obviously closed in KN. Thus the bounded and closed
+set BFν(N) is compact in F(N), implying our statement by Theorem 5.5.1.
+□
+5.5.4. Corollary. Let Ω ⊂ Rd be open, P(∂)K a hypoelliptic linear partial
+differential operator, ν∶Ω → (0,∞) continuous and (xn)n∈N fix the topology in
+CνP (∂)(Ω).
+a) Then there is 0 < λ ∈ ℓ1 such that for every bounded B ⊂ (C∞
+P (∂)(Ω),τc)′
+b
+there is C ≥ 1 with
+{µ∣CνP (∂)(Ω) ∣ µ ∈ B} ⊂ {
+∞
+∑
+n=1
+anν(xn)δxn ∈ CνP (∂)(Ω)′ ∣ a ∈ ℓ1, ∀ n ∈ N ∶ ∣an∣ ≤ Cλn}.
+b) Then there is a decreasing zero sequence (εn)n∈N such that for all compact
+K ⊂ Ω there is C ≥ 1 with
+sup
+x∈K
+∣f(x)∣ ≤ C sup
+n∈N
+∣f(xn)∣ν(xn)εn,
+f ∈ CνP (∂)(Ω).
+Proof. Due to the proof of Corollary 5.2.30 and the observation that the space
+F(Ω) = (C∞
+P (∂)(Ω),τc) is a nuclear Fréchet space all conditions of Theorem 5.5.1
+are fulfilled, which yields our statement.
+□
+5.6. Series representation of vector-valued functions via Schauder
+decompositions
+The purpose of this section is to lift series representations known from scalar-
+valued functions to vector-valued functions and its underlying idea was derived from
+the classical example of the (local) power series representation of a holomorphic
+function. We recall that a C-valued function f on the open disc Dr(0) around zero
+
+118
+5. APPLICATIONS
+with radius r > 0 belongs to the space O(Dr(0)) of holomorphic functions on Dr(0)
+if the limit
+f (1)(z) ∶=
+lim
+h→0
+h∈C,h≠0
+f(z + h) − f(z)
+h
+,
+z ∈ Dr(0),
+(58)
+exists in C. It is well-known that every f ∈ O(Dr(0)) can be written as
+f(z) =
+∞
+∑
+n=0
+f (n)(0)
+n!
+zn,
+z ∈ Dr(0),
+where the power series on the right-hand side converges uniformly on every com-
+pact subset of Dr(0) and f (n)(0) is the n-th complex derivative of f at 0 which
+is defined from (58) by the recursion f (0) ∶= f and f (n) ∶= (f (n−1))(1) for n ∈ N.
+By [79, 2.1 Theorem and Definition, p. 17–18] and [79, 5.2 Theorem, p. 35], this
+series representation remains valid if f is a holomorphic function on Dr(0) with
+values in a locally complete locally convex Hausdorff space E over C where holo-
+morphy means that the limit (58) exists in E and the higher complex derivatives
+are defined recursively as well. Analysing this example, we observe that O(Dr(0)),
+equipped with the topology τc of uniform convergence on compact subsets of Dr(0),
+is a Fréchet space, in particular, barrelled, with a Schauder basis formed by the
+monomials z ↦ zn. Further, the formulas for the complex derivatives of a C-valued
+resp. an E-valued function f on Dr(0) are built up in the same way by (58) (see
+Chapter 2).
+Our goal is to derive a mechanism which uses these observations and transfers
+known series representations for other spaces of scalar-valued functions to their
+vector-valued counterparts. Let us describe the general setting. We recall from [89,
+14.2, p. 292] that a sequence (fn) in a locally convex Hausdorff space F over a field
+K is called a topological basis, or simply a basis, if for every f ∈ F there is a unique
+sequence of coefficients (λK
+n(f)) in K such that
+f =
+∞
+∑
+n=1
+λK
+n(f)fn
+(59)
+where the series converges in F.
+Due to the uniqueness of the coefficients the
+map λK
+n∶f ↦ λK
+n(f) is well-defined, linear and called the n-th coefficient functional
+associated to (fn). Further, for each k ∈ N the map
+Pk∶F → F, Pk(f) ∶=
+k
+∑
+n=1
+λK
+n(f)fn,
+is a linear projection whose range is span{f1,...,fn} and it is called the k-th ex-
+pansion operator associated to (fn). A basis (fn) of F is called equicontinuous if
+the expansion operators Pk form an equicontinuous sequence in the linear space
+L(F,F) of continuous linear maps from F to F (see [89, 14.3, p. 296]). A basis
+(fn) of F is called a Schauder basis if the coefficient functionals are continuous, i.e.
+λK
+n ∈ F ′ for each n ∈ N. In particular, this is already fulfilled if F is a Fréchet space
+by [131, Corollary 28.11, p. 351]. If F is barrelled, then a Schauder basis of F is
+already equicontinuous and F has the (bounded) approximation property by the
+uniform boundedness principle.
+The starting point for our approach is equation (59). Let F and E be non-
+trivial locally convex Hausdorff spaces over a field K where F has an equicontinuous
+Schauder basis (fn) with associated coefficient functionals (λK
+n). The expansion
+operators (Pk) form a so-called Schauder decomposition of F (see [27, p. 77]), i.e.
+they are continuous projections on F such that
+(i) PkPj = Pmin(j,k) for all j,k ∈ N,
+(ii) Pk ≠ Pj for k ≠ j,
+
+5.6. SERIES REPRESENTATION OF VECTOR-VALUED FUNCTIONS
+119
+(iii) (Pkf) converges to f for each f ∈ F.
+This operator theoretic definition of a Schauder decomposition is equivalent to the
+usual definition in terms of closed subspaces of F given in [96, p. 377] (see [123,
+p. 219]). In our main Theorem 5.6.1 of this section we prove that (PkεidE) is a
+Schauder decomposition of Schwartz’ ε-product FεE and each u ∈ FεE has the
+series representation
+u(f ′) =
+∞
+∑
+n=1
+u(λK
+n)f ′(fn),
+f ′ ∈ F ′.
+Now, suppose that F = F(Ω) is a space of K-valued functions on a set Ω with a
+topology such that the point-evaluation functionals δx, x ∈ Ω, belong to F(Ω)′ and
+that there is a locally convex Hausdorff space F(Ω,E) of functions from Ω to E
+such that the map
+S∶F(Ω)εE → F(Ω,E), u �→ [x ↦ u(δx)],
+is an isomorphism, i.e. suppose that F(Ω) and F(Ω,E) are ε-compatible. Assuming
+that for each n ∈ N and u ∈ F(Ω)εE there is λE
+n (S(u)) ∈ E with
+λE
+n (S(u)) = u(λK
+n),
+(60)
+i.e. (λE,λK) is consistent, we obtain in Corollary 5.6.5 that (S ○ (PkεidE) ○ S−1)k
+is a Schauder decomposition of F(Ω,E) and
+f = lim
+k→∞(S ○ (PkεidE) ○ S−1)(f) =
+∞
+∑
+n=1
+λE
+n (f)fn,
+f ∈ F(Ω,E),
+which is the desired series representation in F(Ω,E). In particular, the consis-
+tency condition (60) guarantees in the case of E-valued holomorphic functions on
+Dr(0) that the complex derivatives at 0 appear in the Schauder decomposition of
+O(Dr(0),E) since (∂n
+C)ES(u)(0) = u(δ0 ○(∂n
+C)C) for all u ∈ O(Dr(0))εE and n ∈ N0
+by Proposition 5.2.32 if E is locally complete. We apply our result to sequence
+spaces, spaces of continuously differentiable functions on a compact interval, the
+space of holomorphic functions, the Schwartz space and the space of smooth func-
+tions which are 2π-periodic in each variable.
+As a byproduct of Theorem 5.6.1 we obtain that every element of the completion
+F ̂⊗εE of the injective tensor product F ⊗ε E has a series representation as well if
+F is a complete space with an equicontinuous Schauder basis and E is complete.
+Concerning series representation in F ̂⊗εE, little seems to be known whereas for
+the completion F ̂⊗πE of the projective tensor product F ⊗π E of two metrisable
+locally convex spaces F and E it is well-known that every f ∈ F ̂⊗πE has a series
+representation
+f =
+∞
+∑
+n=1
+anfn ⊗ en
+where (an) ∈ ℓ1, i.e. (an) is absolutely summable, and (fn) and (en) are null
+sequences in F and E, respectively (see e.g. [83, Chap. I, §2 , n○1, Théorème 1, p.
+51] or [89, 15.6.4 Corollary, p. 334]). If F and E are metrisable and one of them is
+nuclear, then the isomorphism F ̂⊗πE ≅ F ̂⊗εE holds and we trivially have a series
+representation of the elements of F ̂⊗εE as well. Other conditions on the existence
+of series representations of the elements of F ̂⊗εE can be found in [151, Proposition
+4.25, p. 88], where F and E are Banach spaces and both of them have a Schauder
+basis, and in [91, Theorem 2, p. 283], where F and E are locally convex Hausdorff
+spaces and both of them have an equicontinuous Schauder basis.
+
+120
+5. APPLICATIONS
+5.6.1. Schauder decomposition. Let us start with our main theorem on
+Schauder decompositions of ε-products. We recall from (3) that we consider the
+tensor product F ⊗ E as a linear subspace of FεE for two locally convex Hausdorff
+spaces F and E by means of the linear injection
+Θ∶F ⊗ E → FεE,
+k
+∑
+n=1
+fn ⊗ en �→ [y ↦
+k
+∑
+n=1
+y(fn)en].
+The next theorem is essentially due to José Bonet, improving a previous version
+of us which became Corollary 5.6.5.
+5.6.1. Theorem. Let F and E be lcHs, (fn)n∈N an equicontinuous Schauder
+basis of F with associated coefficient functionals (λn)n∈N and set Qn∶F → F,
+Qn(f) ∶= λn(f)fn for every n ∈ N. Then the following holds:
+a) The sequence (Pk)k∈N given by Pk ∶= (∑k
+n=1 Qn)εidE is a Schauder decom-
+position of FεE.
+b) Each u ∈ FεE has the series representation
+u(f ′) =
+∞
+∑
+n=1
+u(λn)f ′(fn),
+f ′ ∈ F ′.
+c) F ⊗ E is sequentially dense in FεE.
+Proof. Since (fn) is a Schauder basis of F, the sequence (∑k
+n=1 Qn) converges
+to idF in Lσ(F,F). Thus we deduce from the equicontinuity of (fn) that (∑k
+n=1 Qn)
+converges to idF in Lκ(F,F) by [89, Theorem 8.5.1 (b), p. 156]. For f ′ ∈ F ′ and
+f ∈ F it holds
+(Qt
+n ○ Qt
+m)(f ′)(f) = Qt
+m(f ′)(Qn(f)) = Qt
+m(f ′)(λn(f)fn) = f ′(λm(λn(f)fn)fm)
+= λm(fn)λn(f)f ′(fm) =
+⎧⎪⎪⎨⎪⎪⎩
+λn(f)f ′(fn)
+, m = n,
+0
+, m ≠ n,
+due to the uniqueness of the coefficient functionals (λn) (see [89, 14.2.1 Proposition,
+p. 292]) and it follows for k,j ∈ N that
+(
+j
+∑
+n=1
+Qt
+n ○
+k
+∑
+m=1
+Qt
+m)(f ′)(f) =
+min(j,k)
+∑
+n=1
+λn(f)f ′(fn) =
+min(j,k)
+∑
+n=1
+Qt
+n(f ′)(f).
+This implies that
+(PkPj)(u) = u ○
+j
+∑
+n=1
+Qt
+n ○
+k
+∑
+m=1
+Qt
+m = u ○
+min(j,k)
+∑
+n=1
+Qt
+n = Pmin(j,k)(u)
+for all u ∈ FεE. If k ≠ j, w.l.o.g. k > j, we choose x ∈ E, x ≠ 0,3 and consider fk ⊗ x
+as an element of FεE via the map Θ. Then
+(Pk − Pj)(fk ⊗ x) =
+k
+∑
+n=j+1
+(fk ⊗ x) ○ Qt
+n = fk ⊗ x ≠ 0
+since
+((fk⊗x)○Qt
+n)(f ′) = (fk⊗x)(λn(⋅)f ′(fn)) = λn(fk)f ′(fn)x =
+⎧⎪⎪⎨⎪⎪⎩
+(fk ⊗ x)(f ′) , n = k,
+0
+, n ≠ k.
+It remains to prove that for each u ∈ FεE
+lim
+k→∞Pk(u) = u
+3The lcHs E is non-trivial by our assumptions in Chapter 2.
+
+5.6. SERIES REPRESENTATION OF VECTOR-VALUED FUNCTIONS
+121
+in FεE. Let (qβ)β∈B denote the system of seminorms inducing the locally convex
+topology of F. Let u ∈ FεE and α ∈ A. Due to the continuity of u there are an
+absolutely convex compact set K = K(u,α) ⊂ F and C0 = C0(u,α) > 0 such that
+for each f ′ ∈ F ′ we have
+pα((Pk(u) − u)(f ′)) = pα(u((
+k
+∑
+n=1
+Qt
+n − idF ′)(f ′))) ≤ C0 sup
+f∈K
+∣(
+k
+∑
+n=1
+Qt
+n − idF ′)(f ′)(f)∣
+= C0 sup
+f∈K
+∣f ′(
+k
+∑
+n=1
+Qnf − f)∣.
+Let V be an absolutely convex zero neighbourhood in F. As a consequence of the
+equicontinuity of the polar V ○ there are C1 > 0 and β ∈ B such that
+sup
+f ′∈V ○ pα((Pk(u) − u)(f ′)) ≤ C0C1 sup
+f∈K
+qβ(
+k
+∑
+n=1
+Qnf − f).
+In combination with the convergence of (∑k
+n=1 Qn) to idF in Lκ(F,F) this yields
+the convergence of (Pk(u)) to u in FεE and settles part a).
+Let us turn to b) and c). Since
+Pk(u)(f ′) = u(
+k
+∑
+n=1
+Qt
+n(f ′)) =
+k
+∑
+n=1
+u(λn)f ′(fn)
+for every f ′ ∈ F ′, we note that the range of Pk(u) is contained in span{u(λn) ∣ 1 ≤
+n ≤ k} for each u ∈ FεE and k ∈ N. Hence Pk(u) has finite rank and thus belongs
+to F ⊗ E, implying the sequential density of F ⊗ E in FεE and the desired series
+representation by part a).
+□
+The index set of the equicontinuous Schauder basis of F in Theorem 5.6.1 need
+not be N (or N0) but may be any other countable index set as long as the equicon-
+tinuous Schauder basis is unconditional which is, for instance, always fulfilled if F
+is nuclear by [89, 21.10.1 Dynin-Mitiagin Theorem, p. 510].
+5.6.2. Remark. If F and E are complete, we have under the assumption of
+Theorem 5.6.1 that F ̂⊗εE ≅ FεE by c) since FεE is complete by [94, Satz 10.3, p.
+234] and F ̂⊗εE is the closure of F ⊗ E in FεE. Thus each element of F ̂⊗εE has
+a series representation.
+Let us apply the preceding theorem to spaces of Lebesgue integrable functions.
+We consider the measure space ([0,1],L ([0,1]),λ) of Lebesgue measurable sets
+and use the notation Lp[0,1] for the space of (equivalence classes) of Lebesgue
+p-integrable functions on [0,1]. The Haar system hn∶[0,1] → R, n ∈ N, given by
+h1(x) ∶= 1 for all x ∈ [0,1] and
+h2k+j(x) ∶=
+⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
+1
+,(2j − 2)/2k+1 ≤ x < (2j − 1)/2k+1,
+−1
+,(2j − 1)/2k+1 ≤ x < 2j/2k+1,
+0
+,else,
+for k ∈ N0 and 1 ≤ j ≤ 2k forms a Schauder basis of Lp[0,1] for every 1 ≤ p < ∞ and
+the associated coefficient functionals are
+λn(f) ∶= ∫
+[0,1]
+f(x)hn(x)dλ(x),
+f ∈ Lp[0,1], n ∈ N,
+(see [154, Satz I, p. 317]). Because Lp[0,1] is Banach space and thus barrelled, its
+Schauder basis (hn) is equicontinuous and we directly obtain from Theorem 5.6.1
+the following corollary.
+
+122
+5. APPLICATIONS
+5.6.3. Corollary. Let E be an lcHs and 1 ≤ p < ∞. (∑k
+n=1 λn(⋅)hnεidE)k∈N is
+a Schauder decomposition of Lp[0,1]εE and for each u ∈ Lp[0,1]εE it holds
+u(f ′) =
+∞
+∑
+n=1
+u(λn)f ′(hn),
+f ′ ∈ Lp[0,1]′.
+Defining Lp([0,1],E) ∶= Lp[0,1]εE, we can read the corollary above as a state-
+ment on series representations in the vector-valued version of Lp[0,1]. However,
+in many cases of spaces F(Ω) of scalar-valued functions there is a more natural
+way to define the vector-valued version F(Ω,E) of F(Ω), namely, that F(Ω) and
+F(Ω,E) are ε-compatible.
+5.6.4. Remark. If F(Ω) and F(Ω,E) are ε-into-compatible, then we get by
+identification of isomorphic subspaces
+F(Ω) ⊗ε E ⊂ F(Ω)εE ⊂ F(Ω,E)
+and the embedding F(Ω) ⊗ E ↪ F(Ω,E) is given by f ⊗ e �→ [x ↦ f(x)e].
+Proof. The inclusions obviously hold and F(Ω)εE and F(Ω,E) induce the
+same topology on F(Ω) ⊗ E. Further, we have
+f ⊗ e
+Θ
+�→ [y ↦ y(f)e]
+S
+�→ [x �→ [y ↦ y(f)e](δx)] = [x ↦ f(x)e].
+□
+5.6.5. Corollary. Let F(Ω) and F(Ω,E) be ε-compatible, (fn)n∈N an equi-
+continuous Schauder basis of F(Ω) with associated coefficient functionals λK ∶=
+(λK
+n)n∈N. Let there be λE∶F(Ω,E) → EN such that (λE,λK) is a consistent family
+for (F,E), and set QE
+n ∶F(Ω,E) → F(Ω,E), QE
+n (f) ∶= λE
+n (f)fn for every n ∈ N.
+Then the following holds:
+a) The sequence (P E
+k )k∈N given by P E
+k ∶= ∑k
+n=1 QE
+n is a Schauder decomposi-
+tion of F(Ω,E).
+b) Each f ∈ F(Ω,E) has the series representation
+f =
+∞
+∑
+n=1
+λE
+n (f)fn.
+c) F(Ω) ⊗ E is sequentially dense in F(Ω,E).
+Proof. For each u ∈ F(Ω)εE and x ∈ Ω we note that with Pk from Theorem
+5.6.1 it holds
+(S ○ Pk)(u)(x) = u(
+k
+∑
+n=1
+Qt
+n(δx)) = u(
+k
+∑
+n=1
+λK
+n(⋅)fn(x)) =
+k
+∑
+n=1
+u(λK
+n)fn(x)
+=
+k
+∑
+n=1
+λE
+n (S(u))fn(x) = (P E
+k ○ S)(u)(x),
+which means that S ○ Pk = P E
+k ○ S. This implies part a) and b) by Theorem 5.6.1
+a) since S is an isomorphism. Part c) is a direct consequence of Theorem 5.6.1 c)
+and the isomorphism F(Ω)εE ≅ F(Ω,E).
+□
+In the preceding corollary we used the isomorphism S to obtain a Schauder
+decomposition. On the other hand, if S is an isomorphism into, which is often the
+case (see Theorem 3.1.12), we can use a Schauder decomposition of F(Ω,E) to
+prove the surjectivity of S.
+5.6.6. Proposition. Let F(Ω) and F(Ω,E) be ε-into-compatible. Let there be
+(fn)n∈N in F(Ω) and for every f ∈ F(Ω,E) a sequence (λE
+n (f))n∈N in E such that
+f =
+∞
+∑
+n=1
+λE
+n (f)fn,
+f ∈ F(Ω,E).
+Then the following holds:
+
+5.6. SERIES REPRESENTATION OF VECTOR-VALUED FUNCTIONS
+123
+a) F(Ω) ⊗ E is sequentially dense in F(Ω,E).
+b) If F(Ω) and E are sequentially complete, then
+F(Ω,E) ≅ F(Ω)εE.
+c) If F(Ω) and E are complete, then
+F(Ω,E) ≅ F(Ω)εE ≅ F(Ω)̂⊗εE.
+Proof. Let f ∈ F(Ω,E) and observe that
+P E
+k (f) ∶=
+k
+∑
+n=1
+λE
+n (f)fn =
+k
+∑
+n=1
+fn ⊗ λE
+n (f) ∈ F(Ω) ⊗ E
+for every k ∈ N by Remark 5.6.4. Due to our assumption we have the convergence
+P E
+k (f) → f in F(Ω,E). Thus F(Ω) ⊗ E is sequentially dense in F(Ω,E).
+Let us turn to part b). If F(Ω) and E are sequentially complete, then F(Ω)εE
+is sequentially complete by [94, Satz 10.3, p. 234]. Since S is an isomorphism into
+and
+S(Θ(
+k
+∑
+n=q
+fn ⊗ λE
+n (f))) =
+k
+∑
+n=q
+λE
+n (f)fn
+for all k,q ∈ N, k > q, we get that (Θ(∑k
+n=1 fn ⊗ λE
+n (f)) is a Cauchy sequence in
+F(Ω)εE and thus convergent. Hence we deduce that
+S( lim
+k→∞Θ(
+k
+∑
+n=1
+fn ⊗ λE
+n (f))) = lim
+k→∞
+k
+∑
+n=1
+(S ○ Θ)(fn ⊗ λE
+n (f)) =
+∞
+∑
+n=1
+λE
+n (f)fn = f,
+which proves the surjectivity of S.
+If F(Ω) and E are complete, then F(Ω)̂⊗εE is the closure of F(Ω)⊗ε E in the
+complete space F(Ω)εE by [94, Satz 10.3, p. 234]. As limk→∞ Θ(∑k
+n=1 fn ⊗ λE
+n (f))
+is an element of the closure, we obtain part c).
+□
+5.6.2. Examples of Schauder decompositions.
+Sequence spaces. For our first application we recall the definition of some
+sequence spaces. For an lcHs E and a Köthe matrix A ∶= (ak,j)k,j∈N we define the
+topological subspace of λ∞(A,E) from Corollary 4.2.3 a) by
+c0(A,E) ∶= {x = (xk) ∈ EN ∣ ∀ j ∈ N ∶ lim
+k→∞xkak,j = 0}.
+In particular, the space c0(N,E) of null-sequences in E is obtained as c0(N,E) =
+c0(A,E) with ak,j ∶= 1 for all k,j ∈ N. The space of convergent sequences in E is
+defined by
+c(N,E) ∶= {x ∈ EN ∣ x = (xk) converges in E}
+and equipped with the system of seminorms
+∣x∣α ∶= sup
+k∈N
+pα(xk),
+x ∈ c(N,E),
+for α ∈ A. Further, we set c0(A) ∶= c0(A,K), c0(N) ∶= c0(N,K) and c(N) ∶= c(N,K).
+Furthermore, we equip the space EN with the system of seminorms given by
+∥x∥l,α ∶= sup
+k∈N
+pα(xk)χ{1,...,l}(k),
+x = (xk) ∈ EN,
+for l ∈ N and α ∈ A. For a non-empty set Ω we define for n ∈ Ω the n-th unit
+function by
+ϕn,Ω∶Ω → K, ϕn,Ω(k) ∶=
+⎧⎪⎪⎨⎪⎪⎩
+1
+, k = n,
+0
+, else,
+and we simply write ϕn instead of ϕn,Ω if no confusion seems to be likely. Further,
+we set ϕ∞∶N → K, ϕ∞(k) ∶= 1, and x∞ ∶= δ∞(x) ∶= limk→∞ xk for x ∈ c(N,E).
+For series representations of the elements in these sequence spaces we do not need
+
+124
+5. APPLICATIONS
+Corollary 5.6.5 due to the subsequent proposition but we can use the representation
+to obtain the surjectivity of S for sequentially complete E.
+5.6.7. Proposition. Let E be an lcHs and ℓ(Ω,E) one of the spaces c0(A,E),
+EN, s(Nd,E), s(Nd
+0,E) or s(Zd,E).
+a) Then (∑n∈Ω,∣n∣≤k δnϕn)k∈N is a Schauder decomposition of ℓ(Ω,E) and
+x = ∑
+n∈Ω
+xnϕn,
+x ∈ ℓ(Ω,E).
+b) Then (δ∞ϕ∞+∑k
+n=1(δn−δ∞)ϕn)k∈N is a Schauder decomposition of c(N,E)
+and
+x = x∞ϕ∞ +
+∞
+∑
+n=1
+(xn − x∞)ϕn,
+x ∈ c(N,E).
+Proof. Let us begin with a). First, we note that (ϕn)n∈Ω is an unconditional
+equicontinuous Schauder basis of s(Ω), Ω = Nd, Nd
+0, Zd, since s(Ω) is a nuclear
+Fréchet space. Now, for x = (xn) ∈ ℓ(Ω,E) let (P E
+k ) be the sequence in ℓ(Ω,E)
+given by P E
+k (x) ∶= ∑∣n∣≤k xnϕn. It is easy to see that P E
+k is a continuous projection
+on ℓ(Ω,E), P E
+k P E
+j
+= P E
+min(k,j) for all k,j ∈ N and P E
+k ≠ P E
+j
+for k ≠ j. Let ε > 0,
+α ∈ A and j ∈ N. For x ∈ c0(A,E) there is N0 ∈ N such that pα(xnan,j) < ε for all
+n ≥ N0. Hence we have for x ∈ c0(A,E)
+∣x − P E
+k (x)∣j,α = sup
+n>k
+pα(xn)an,j ≤ sup
+n≥N0
+pα(xn)an,j ≤ ε
+for all k ≥ N0. For x ∈ EN and l ∈ N we have
+∥x − P E
+k (x)∥l,α = 0 < ε
+for all k ≥ l. For x ∈ s(Ω,E), Ω = Nd, Nd
+0, Zd, we notice that there is N1 ∈ N such
+that for all n ∈ Ω with ∣n∣ ≥ N1 we have
+(1 + ∣n∣2)j/2
+(1 + ∣n∣2)j
+= (1 + ∣n∣2)−j/2 < ε.
+Thus we deduce for ∣n∣ ≥ N1
+pα(xn)(1 + ∣n∣2)j/2 < εpα(xn)(1 + ∣n∣2)j ≤ ε∣x∣2j,α
+and hence
+∣x − P E
+k (x)∣j,α = sup
+∣n∣>k
+pα(xn)(1 + ∣n∣2)j/2 ≤ sup
+∣n∣≥N1
+pα(xn)(1 + ∣n∣2)j/2 ≤ ε∣x∣2j,α
+for all k ≥ N1. Therefore (P E
+k (x)) converges to x in ℓ(Ω,E) and
+x = lim
+k→∞P E
+k (x) = ∑
+n∈Ω
+xnϕn.
+Now, we turn to b). For x = (xn) ∈ c(N,E) let ( ̃P E
+k (x)) be the sequence in
+c(N,E) given by ̃P E
+k (x) ∶= x∞ϕ∞ + ∑k
+n=1(xn − x∞)ϕn.
+Again, it is easy to see
+that ̃P E
+k is a continuous projection on c(N,E), ̃P E
+k ̃P E
+j
+= ̃P E
+min(k,j) for all k,j ∈ N
+and ̃P E
+k ≠ ̃P E
+j
+for k ≠ j. Let ε > 0 and α ∈ A. Then there is N2 ∈ N such that
+pα(xn − x∞) < ε for all n ≥ N2. Thus we obtain
+∣x − ̃P E
+k (x)∣α = sup
+n>k
+pα(xn − x∞) ≤ sup
+n≥N2
+pα(xn − x∞) ≤ ε
+for all k ≥ N2, implying that ( ̃P E
+k (x)) converges to x in c(N,E) and
+x = lim
+k→∞
+̃P E
+k (x) = x∞ϕ∞ +
+∞
+∑
+n=1
+(xn − x∞)ϕn.
+□
+
+5.6. SERIES REPRESENTATION OF VECTOR-VALUED FUNCTIONS
+125
+5.6.8. Theorem. Let E be a sequentially complete lcHs and ℓ(Ω,E) one of the
+spaces c0(A,E), EN, s(Nd,E), s(Nd
+0,E) or s(Zd,E). Then
+(i) ℓ(Ω,E) ≅ ℓ(Ω)εE,
+(ii) c(N,E) ≅ c(N)εE.
+Proof. The map Sℓ(Ω) is an isomorphism into by Theorem 3.1.12 and, in
+addition, by Proposition 4.1.9 (i) if ℓ(Ω,E) = c0(A,E). Considering c(N,E), we
+observe that for x ∈ c(N)
+δn(x) = xn → x∞ = δ∞(x),
+which implies the convergence δn → δ∞ in c(N)′
+γ by the Banach–Steinhaus theorem
+since c(N) is a Banach space. Hence we get
+u(δ∞) = lim
+n→∞u(δn) = lim
+n→∞S(u)(n) = δ∞(S(u))
+for every u ∈ c(N)εE, which implies that Sc(N) is an isomorphism into by Theorem
+3.1.12. From Proposition 5.6.7 and Proposition 5.6.6 we deduce our statement.
+□
+More general, we note that Theorem 5.6.8 holds for any lcHs E if ℓ(Ω,E) = EN
+by Example 4.2.1, for E with metric ccp if ℓ(Ω,E) = c0(A,E) by Example 4.2.11 (ii),
+and for locally complete E if ℓ(Ω,E) = s(Ω,E) with Ω = Nd, Nd
+0, Zd by Corollary
+4.2.3 b).
+Continuous and differentiable functions on a compact interval. We
+start with continuous functions on compact sets. Let E be an lcHs and Ω ⊂ Rd
+compact. We equip the space C(Ω,E) of continuous functions on Ω with values in
+E with the system of seminorms given by
+∣f∣α ∶= sup
+x∈Ω
+pα(f(x)),
+f ∈ C(Ω,E),
+for α ∈ A. We want to apply our preceding results to intervals. Let −∞ < a < b < ∞
+and T ∶= (tj)0≤j≤n be a partition of the interval [a,b], i.e. a = t0 < t1 < ... < tn = b.
+The hat functions hT
+tj∶[a,b] → R for the partition T are given by
+hT
+tj(x) ∶=
+⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
+x−tj
+tj−tj−1
+,tj−1 ≤ x ≤ tj,
+tj+1−x
+tj+1−tj
+,tj < x ≤ tj+1,
+0
+,else,
+for 2 ≤ j ≤ n − 1 and
+hT
+a (x) ∶=
+⎧⎪⎪⎨⎪⎪⎩
+t1−x
+t1−a
+,a ≤ x ≤ t1,
+0
+,else,
+hT
+b (x) ∶=
+⎧⎪⎪⎨⎪⎪⎩
+x−tn−1
+b−tn−1
+,tn−1 ≤ x ≤ b,
+0
+,else.
+Let T ∶= (tn)n∈N0 be a dense sequence in [a,b] with t0 = a, t1 = b and tn ≠ tm for n ≠
+m. For T n ∶= {t0,...,tn} there is a (unique) enumeration σ∶{0,...,n} → {0,...,n}
+of T n such that Tn ∶= (tσ(j))0≤j≤n is a partition of [a,b] with T n = {tσ(1),...,tσ(n)}.
+The functions ϕT
+0 ∶= hT1
+t0 , ϕT
+1 ∶= hT1
+t1 and ϕT
+n ∶= hTn
+tσ(j) with j = σ−1(n) for n ≥ 2
+are called Schauder hat functions for the sequence T and form a Schauder basis of
+C([a,b]) with associated coefficient functionals given by λK
+0 (f) ∶= f(t0), λK
+1 (f) ∶=
+f(t1) and
+λK
+n+1(f) ∶= f(tn+1) −
+n
+∑
+k=0
+λK
+k (f)ϕT
+k (tn+1),
+f ∈ C([a,b]), n ≥ 1,
+by [166, 2.3.5 Proposition, p. 29]. Looking at the coefficient functionals, we see that
+the right-hand sides even make sense if f ∈ C([a,b],E) and thus we define λE
+n on
+C([a,b],E) for n ∈ N0 accordingly.
+
+126
+5. APPLICATIONS
+5.6.9. Theorem. Let E be an lcHs with metric ccp and T ∶= (tn)n∈N0 a dense
+sequence in [a,b] with t0 = a, t1 = b and tn ≠ tm for n ≠ m. Then (∑k
+n=0 λE
+k ϕT
+n )k∈N0
+is a Schauder decomposition of C([a,b],E) and
+f =
+∞
+∑
+n=0
+λE
+n (f)ϕT
+n ,
+f ∈ C([a,b],E).
+Proof. The spaces C([a,b]) and C([a,b],E) are ε-compatible by Example
+4.2.12 if E has metric ccp. C([a,b]) is a Banach space and thus barrelled, implying
+that its Schauder basis (ϕT
+n ) is equicontinuous. We note that for all u ∈ C([a,b])εE
+and x ∈ [a,b]
+λE
+n (S(u))(x) = u(δtn) = u(λK
+n),
+n ∈ {0,1},
+and by induction
+λE
+n+1(S(u))(x) = u(δtn+1) −
+n
+∑
+k=0
+λE
+k (S(u))ϕT
+k (tn+1) = u(δtn+1) −
+n
+∑
+k=0
+u(λK
+k )ϕT
+k (tn+1)
+= u(λK
+n+1),
+n ≥ 1.
+Thus (λE,λK) is consistent, proving our claim by Corollary 5.6.5.
+□
+If a = 0, b = 1 and T is the sequence of dyadic numbers given in [166, 2.1.1
+Definitions, p. 21], then (ϕT
+n ) is the so-called Faber–Schauder system. Using the
+Schauder basis and coefficient functionals of the space C0(R) of continuous functions
+vanishing at infinity given in [166, 2.7.1, p. 41–42] and [166, 2.7.4 Corollary, p. 43]
+and that SC0(R) is an isomorphism by Example 4.2.11 (ii) if E has metric ccp,
+the corresponding result for the E-valued counterpart C0(R,E) holds as well by a
+similar reasoning. Another corresponding result holds for the space C[γ]
+0,0 ([0,1],E),
+0 < γ < 1, of γ-Hölder continuous functions on [0,1] with values in E that vanish
+at zero and at infinity if one uses the Schauder basis and coefficient functionals of
+C[γ]
+0,0 ([0,1]) from [44, Theorem 2, p. 220] and [43, Theorem 3, p. 230]. This result is
+a bit weaker since Example 4.2.9 only guarantees that SC[γ]
+0,0([0,1]) is an isomorphism
+if E is quasi-complete.
+Now, we turn to the spaces Ck([a,b],E) of continuously differentiable functions
+on an interval (a,b) with values in an lcHs E such that all derivatives can be
+continuously extended to the boundary from Example 4.2.28. We set f (k)(x) ∶=
+(∂k)Kf(x) for x ∈ (a,b) and f ∈ Ck([a,b]). From the Schauder hat functions (ϕT
+n )
+for a dense sequence T ∶= (tn)n∈N0 in [a,b] with t0 = a, t1 = b and tn ≠ tm for n ≠ m
+and the associated coefficient functionals λK
+n we can easily get a Schauder basis for
+the space Ck([a,b]), k ∈ N, by applying ∫
+(⋅)
+a
+k-times to the series representation
+f (k) =
+∞
+∑
+n=0
+λK
+n(f (k))ϕT
+n ,
+f ∈ Ck([a,b]),
+where we identified f (k) with its continuous extension. The resulting Schauder basis
+f T
+n ∶[a,b] → R and associated coefficient functionals µK
+n∶Ck([a,b]) → K, n ∈ N0, are
+f T
+n (x) = 1
+n!(x − a)n,
+µK
+n(f) = f (n)(a),
+0 ≤ n ≤ k − 1,
+f T
+n (x) =
+x
+∫
+a
+sk−1
+∫
+a
+⋯
+s2
+∫
+a
+s1
+∫
+a
+ϕT
+n−kdsds1 ...dsk−1,
+µK
+n(f) = λK
+n−k(f (k)),
+n ≥ k,
+for x ∈ [a,b] and f ∈ Ck([a,b]) (see e.g. [157, p. 586–587], [166, 2.3.7, p. 29]). Again,
+the mapping rule for the coefficient functionals still makes sense if f ∈ Ck([a,b],E)
+and so we define µE
+n on Ck([a,b],E) for n ∈ N0 accordingly.
+
+5.6. SERIES REPRESENTATION OF VECTOR-VALUED FUNCTIONS
+127
+5.6.10. Theorem. Let E be an lcHs with metric ccp, k ∈ N, T ∶= (tn)n∈N0
+a dense sequence in [a,b] with t0 = a, t1 = b and tn ≠ tm for n ≠ m.
+Then
+(∑l
+n=0 µE
+n f T
+n )l∈N0 is a Schauder decomposition of Ck([a,b],E) and
+f =
+∞
+∑
+n=0
+µE
+n (f)f T
+n ,
+f ∈ Ck([a,b],E).
+Proof. The spaces Ck([a,b]) and Ck([a,b],E) are ε-compatible by Example
+4.2.28 if E has metric ccp. The Banach space Ck([a,b]) is barrelled giving the
+equicontinuity of its Schauder basis. Due to Proposition 3.1.11 c) we have for all
+u ∈ Ck([a,b])εE, β ∈ N0, β ≤ k, and x ∈ (a,b)
+(∂β)ES(u)(x) = u(δx ○ (∂β)K).
+Further, for every sequence (xn) in (a,b) converging to t ∈ {a,b} we obtain by
+Proposition 4.1.7 in combination with Lemma 4.1.8 applied to T ∶= (∂β)K
+lim
+n→∞(∂β)ES(u)(xn) = u( lim
+n→∞δxn ○ (∂β)K).
+From these observations we deduce that µE
+n (S(u)) = u(µK
+n) for all n ∈ N0, i.e.
+(µE,µK) is consistent. Therefore our statement is a consequence of Corollary 5.6.5.
+□
+Holomorphic functions. In this short subsection we show how to get the
+result on power series expansion of holomorphic functions from the introduction.
+Let E be an lcHs over C, z0 ∈ C, r ∈ (0,∞] and equip O(Dr(z0),E) with the
+topology τc of compact convergence.
+5.6.11. Theorem. Let E be a locally complete lcHs over C, z0 ∈ C and r ∈
+(0,∞]. Then (f ↦ ∑k
+n=0
+(∂n
+C )Ef(z0)
+n!
+(⋅ − z0)n)k∈N0 is a Schauder decomposition of
+O(Dr(z0),E) and
+f =
+∞
+∑
+n=0
+(∂n
+C)Ef(z0)
+n!
+(⋅ − z0)n,
+f ∈ O(Dr(z0),E).
+Proof. The spaces O(Dr(z0)) and O(Dr(z0),E) are ε-compatible by Propo-
+sition 4.2.17 and (23) (cf. [30, Theorem 9, p. 232]) if E is locally complete. Further,
+the Schauder basis ((⋅−z0)n) of O(Dr(z0)) is equicontinuous since the Fréchet space
+O(Dr(z0)) is barrelled. Due to Proposition 5.2.32 we have for all u ∈ O(Dr(z0))εE
+(∂n
+C)ES(u)(z) = u(δz ○ (∂n
+C)C),
+n ∈ N0, z ∈ Dr(z0),
+which yields that (λE,λC) is consistent where λE∶O(Dr(z0),E) → EN0 is given
+by λE
+n (f) ∶= (∂n
+C )Ef(z0)
+n!
+for n ∈ N0 (and analogously for E replaced by C). Hence
+Corollary 5.6.5 implies our statement.
+□
+Theorem 5.6.11 holds for holomorphic functions in several variables as well (see
+[113, Theorem 5.7, p. 264]).
+Fourier expansions. In this subsection we turn our attention to Fourier ex-
+pansions in the Schwartz space S(Rd,E) and in the space C∞
+2π(Rd,E) of smooth
+functions that are 2π-periodic in each variable.
+We recall the definition of the Hermite functions. For n ∈ N0 we set
+hn∶R → R, hn(x) ∶= (2nn!√π)−1/2(x − d
+dx)
+n
+e−x2/2 = (2nn!√π)−1/2Hn(x)e−x2/2,
+with the Hermite polynomials Hn of degree n which can be computed recursively
+by
+H0(x) = 1, Hn+1(x) = 2xHn(x) − H′
+n(x) and H′
+n(x) = 2nHn−1(x),
+x ∈ R, n ∈ N0.
+
+128
+5. APPLICATIONS
+For n = (nk) ∈ Nd
+0 we define the n-th Hermite function by
+hn∶Rd → R, hn(x) ∶=
+d
+∏
+k=1
+hnk(xk),
+and
+Hn∶Rd → R, Hn(x) ∶=
+d
+∏
+k=1
+Hnk(xk).
+5.6.12. Proposition. Let E be a locally complete lcHs, f ∈ S(Rd,E) and n ∈
+Nd
+0. Then fhn is Pettis-integrable on Rd.
+Proof. First, we set ψ∶Rd → R, ψ(x) ∶= e−∣x∣2/2, as well as g∶Rd → [0,∞),
+g(x) ∶= e∣x∣2/2. Then ψ ∈ L1(Rd,λ) and ψg = 1. Moreover, let u∶Rd → E, u(x) ∶=
+f(x)hn(x)g(x), and note that
+(∂ej)Eu(x) = (∂ej)Ef(x)hn(x)g(x) + f(x)g(x)∂ejhn(x) + f(x)hn(x)g(x)xj
+where
+∂ejhn(x) = (2njnj!√π)−1/2(H′
+nj(xj)e−x2
+j/2 − Hnj(xj)xje−x2
+j/2)
+d
+∏
+k=1,k≠j
+hnk(xk)
+= (2njnj!√π)−1/2(2njHnj−1(xj) − xjHnj(xj))e−x2
+j/2
+d
+∏
+k=1,k≠j
+hnk(xk)
+for all x = (xk) ∈ Rd and 1 ≤ j ≤ d. We set Cn ∶= (∏d
+i=1 2nini!√π)−1/2 and observe
+that
+g(x)∂ejhn(x) = e∣x∣2/2∂ejhn(x) = Cn(2njHnj−1(xj) − xjHnj(xj))
+d
+∏
+k=1,k≠j
+Hnk(xk)
+is a polynomial in d variables. The functions given by
+hn(x)g(x) = e∣x∣2/2hn(x) = CnHn(x)
+and
+hn(x)g(x)xj = CnHn(x)xj
+are polynomials in d variables as well. Thus there are m ∈ N and C > 0 such that
+max(∣hn(x)g(x)∣,∣g(x)∂ejhn(x)∣,∣hn(x)g(x)xj∣) ≤ C(1 + ∣x∣2)m/2
+for all x ∈ Rd and 1 ≤ j ≤ d, which implies
+pα((∂ej)Eu(x)) ≤ C(pα((∂ej)Ef(x))(1 + ∣x∣2)m/2 + 2pα(f(x))(1 + ∣x∣2)m/2)
+for all α ∈ A and hence
+sup
+x∈Rd
+β∈Nd
+0,∣β∣≤1
+pα((∂β)Eu(x)) ≤ 3C∣f∣S(Rd),m,α.
+Therefore u = fhng is (weakly) C1
+b , which yields u ∈ C[1]
+b (Rd,E) by Proposition
+A.1.5. Further, we set h∶Rd → (0,∞), h(x) ∶= 1 + ∣x∣2, and observe that
+sup
+x∈Rd pα(u(x)h(x)) ≤ sup
+x∈Rd pα(f(x))∣hn(x)g(x)h(x)∣ ≤ C∣f∣m+2,α < ∞
+for all α ∈ A. In addition, we remark that for every ε > 0 there is r > 0 such that
+1 ≤ εh(x) for all x ∉ Br(0) =∶ K. We deduce from Proposition A.2.7 (iii) that fhn
+is Pettis-integrable on Rd.
+□
+Due to the previous proposition we can define the n-th Fourier coefficient of
+f ∈ S(Rd,E) by
+̂f(n) ∶= F E
+n (f) ∶= ∫
+Rd
+f(x)hn(x)dx = ∫
+Rd
+f(x)hn(x)dx,
+n ∈ Nd
+0,
+if E is locally complete. We know that the map
+F K∶S(Rd) → s(Nd
+0), F K(f) ∶= ( ̂f(n))n∈Nd
+0,
+
+5.6. SERIES REPRESENTATION OF VECTOR-VALUED FUNCTIONS
+129
+is an isomorphism (see e.g. [94, Satz 3.7, p. 66]). We improve this result to locally
+complete E and derive a Schauder decomposition of S(Rd,E) as well.
+5.6.13. Theorem. Let E be a locally complete lcHs. Then the following holds:
+a) (∑n∈Nd
+0,∣n∣≤k F E
+n hn)k∈N is a Schauder decomposition of S(Rd,E) and
+f = ∑
+n∈Nd
+0
+̂f(n)hn,
+f ∈ S(Rd,E).
+b) The map
+F E∶S(Rd,E) → s(Nd
+0,E), F E(f) ∶= ( ̂f(n))n∈Nd
+0,
+is an isomorphism and
+F E = Ss(Nd
+0) ○ (F KεidE) ○ S−1
+S(Rd).
+Proof. Let us begin with part a). Due to Corollary 3.2.10 the spaces S(Rd)
+and S(Rd,E) are ε-compatible and the inverse of the isomorphism S∶S(Rd)εE →
+S(Rd,E) is given by the map Rt∶S(Rd,E) → S(Rd)εE, f ↦ J −1 ○ Rt
+f, according
+to Theorem 3.2.4. Moreover, S(Rd) is a nuclear Fréchet space, thus barrelled, and
+hence its Schauder basis (hn) is equicontinuous and unconditional. From the Pettis-
+integrability of fhn by Proposition 5.6.12 and Proposition 4.3.3 with (T E
+0 ,T K
+0 ) ∶=
+(hn idERd ,hn idKRd ) we obtain that (F E,F K) is consistent. Hence we conclude our
+statement from Corollary 5.6.5.
+Let us turn to part b). First, we show that the map F E is well-defined. Let
+f ∈ S(Rd,E). Then e′ ○ f ∈ S(Rd) and
+⟨e′,F E(f)n⟩ = ⟨e′, ̂f(n)⟩ = ̂
+e′ ○ f(n) = F K(e′ ○ f)n
+for every n ∈ Nd
+0 and e′ ∈ E′. Thus we have F K(e′ ○ f) ∈ s(Nd
+0) for every e′ ∈ E′,
+which implies by [131, Mackey’s theorem 23.15, p. 268] that F E(f) ∈ s(Nd
+0,E) and
+that F E is well-defined. Due to Corollary 3.2.10 and Corollary 4.2.3 the maps
+SS(Rd) and Ss(Nd
+0) are isomorphisms, which implies that F E is also an isomorphism
+with F E = Ss(Nd
+0) ○ (F KεidE) ○ S−1
+S(Rd) by Theorem 5.1.2 b).
+□
+Our last example of this subsection is devoted to Fourier expansions in the
+space C∞
+2π(Rd,E). We recall that C∞
+2π(Rd,E) denotes the topological subspace of
+CW∞(Rd,E) consisting of the functions which are 2π-periodic in each variable. Due
+to Lemma A.2.2 we are able to define the n-th Fourier coefficient of f ∈ C∞
+2π(Rd,E)
+by
+̂f(n) ∶= FE
+n (f) ∶= (2π)−d
+∫
+[−π,π]d
+f(x)e−i⟨n,x⟩dx,
+n ∈ Zd,
+where ⟨⋅,⋅⟩ is the usual scalar product on Rd, if E is locally complete. We know
+that the map
+FC∶C∞
+2π(Rd) → s(Zd), FC(f) ∶= ( ̂f(n))n∈Zd,
+is an isomorphism (see e.g. [94, Satz 1.7, p. 18]), which we lift to the E-valued case.
+5.6.14. Theorem. Let E be a locally complete lcHs over C.
+a) Then (∑n∈Zd,∣n∣≤k FE
+n ei⟨n,⋅⟩)k∈N is a Schauder decomposition of C∞
+2π(Rd,E)
+and
+f = ∑
+n∈Zd
+̂f(n)ei⟨n,⋅⟩,
+f ∈ C∞
+2π(Rd,E).
+
+130
+5. APPLICATIONS
+b) The map
+FE∶C∞
+2π(Rd,E) → s(Zd,E), FE(f) ∶= ( ̂f(n))n∈Zd,
+is an isomorphism and
+FE = Ss(Zd) ○ (FCεidE) ○ S−1
+C∞
+2π(Rd).
+Proof. The spaces C∞
+2π(Rd) and C∞
+2π(Rd,E) are ε-compatible by Example
+4.2.27.
+The space C∞
+2π(Rd) is barrelled since it is a nuclear Fréchet space and thus
+its Schauder basis (ei⟨n,⋅⟩) is equicontinuous and unconditional. By Theorem 3.2.4
+the inverse of SC∞
+2π(Rd) is given by Rt∶C∞
+2π(Rd,E) → C∞
+2π(Rd)εE, f ↦ J −1 ○ Rt
+f.
+From the Pettis-integrability of fe−i⟨n,⋅⟩ and Proposition 4.3.3 with (T E
+0 ,T K
+0 ) ∶=
+(e−i⟨n,⋅⟩ idERd ,e−i⟨n,⋅⟩ idCRd ) we obtain that (FE,FC) is consistent. Hence we con-
+clude part a) from Corollary 5.6.5.
+Let us turn to part b). As in Theorem 5.6.13 it follows from [131, Mackey’s
+theorem 23.15, p. 268] that the map FE is well-defined. Due to Corollary 4.2.3 and
+Example 4.2.27 the maps Ss(Zd) and SC∞
+2π(Rd) are isomorphisms, which implies that
+FE is an isomorphism as well with FE = Ss(Zd) ○ (F CεidE) ○ S−1
+C∞
+2π(Rd) by Theorem
+5.1.2 b).
+□
+For quasi-complete E Theorem 5.6.14 is already known by [94, Satz 10.8, p.
+239].
+5.7. Representation by sequence spaces
+Our last section is dedicated to the representation of weighted spaces of E-
+valued functions by weighted spaces of E-valued sequences if there is a counterpart
+of this representation in the scalar-valued case involving the coefficient functionals
+associated to a Schauder basis (see Remark 5.2.3 b)). We only touched upon this
+problem in Section 5.6 for special cases like S(Rd,E) and C∞
+2π(Rd,E) in Theorem
+5.6.13 b) and Theorem 5.6.14 b). We solve this problem in a different way by an
+application of our extension results from Section 5.2. As an example we treat the
+space O(DR(0),E) of holomorphic functions and the multiplier space OM(R,E) of
+the Schwartz space (see Corollary 5.7.3).
+5.7.1. Theorem. Let E be a locally complete lcHs, G ⊂ E′ determine bound-
+edness and F(Ω) and F(Ω,E) resp. ℓ(N) and ℓ(N,E) be ε-into-compatible with
+e′ ○ g ∈ ℓ(N) for all e′ ∈ E′ and g ∈ ℓ(N,E). Let (fn)n∈N be an equicontinuous
+Schauder basis of F(Ω) with associated coefficient functionals (T K
+n )n∈N such that
+T K∶F(Ω) → ℓ(N), T K(f) ∶= (T K
+n (f))n∈N,
+is an isomorphism and let there be T E∶F(Ω,E) → EN such that (T E,T K) is a
+strong, consistent family for (F,E). If
+(i) F(Ω) is a Fréchet–Schwartz space, or
+(ii) E is sequentially complete, G = E′ and F(Ω) is a semi-Montel BC-space,
+then the following holds:
+a) FG(N,E) = ℓ(N,E).
+b) ℓ(N) and ℓ(N,E) are ε-compatible, in particular, ℓ(N)εE ≅ ℓ(N,E).
+c) The map
+T E∶F(Ω,E) → ℓ(N,E), T E(f) ∶= (T E
+n (f))n∈N,
+is a well-defined isomorphism, F(Ω) and F(Ω,E) are ε-compatible, in
+particular, F(Ω)εE ≅ F(Ω,E), and T E = Sℓ(N) ○ (T KεidE) ○ S−1
+F(Ω).
+
+5.7. REPRESENTATION BY SEQUENCE SPACES
+131
+Proof. a)(1) First, we remark that N is a set of uniqueness for (T K,F). Let
+u ∈ F(Ω)εE and n ∈ N. Then
+RN,G(SF(Ω)(u))(n) = (T E ○ SF(Ω))(u)(n) = T E
+n (SF(Ω)(u)) = u(T K
+n ) = u(δn ○ T K)
+= (u ○ (T K)t)(δn) = (T KεidE)(u)(δn)
+= (Sℓ(N) ○ (T KεidE))(u)(n)
+(61)
+by consistency and the ε-into-compatibility, yielding FG(N,E) ⊂ ℓ(N,E) once we
+have shown that RN,G is surjective, which we postpone to part b).
+a)(2) Let g ∈ ℓ(N,E). Then e′ ○g ∈ ℓ(N) for all e′ ∈ E′ and ge′ ∶= (T K)−1(e′ ○g) ∈
+F(Ω). We note that T K
+n (ge′) = (e′ ○ g)(n) for all n ∈ N, which implies ℓ(N,E) ⊂
+FG(N,E).
+b) We only need to show that Sℓ(N) is surjective. Let g ∈ ℓ(N,E), which implies
+g ∈ FG(N,E) by part a)(2).
+We claim that RN,G is surjective. In case (i) this follows directly from Theorem
+5.2.20. Let us turn to case (ii) and denote by (fn)n∈N the equicontinuous Schauder
+basis of F(Ω) associated to (T K
+n )n∈N.
+We check that condition (ii) of Theorem
+5.2.15 is fulfilled. Let f ′ ∈ F(Ω)′ and set
+f ′
+k∶F(Ω) → K, f ′
+k(f) ∶=
+k
+∑
+n=1
+T K
+n (f)f ′(fn),
+for k ∈ N. Then f ′
+k ∈ F(Ω)′ for every k ∈ N and (f ′
+k) converges to f ′ in F(Ω)′
+σ since
+(∑k
+n=1 T K
+n (f)fn) converges to f in F(Ω). From the equicontinuity of the Schauder
+basis we deduce that (f ′
+k) converges to f ′ in F(Ω)′
+κ by [89, 8.5.1 Theorem (b), p.
+156]. Let f ∈ FE′(N,E). For each e′ ∈ E′ and k ∈ N we have
+Rt
+f(f ′
+k)(e′) = f ′
+k(fe′) =
+k
+∑
+n=1
+T K
+n (fe′)f ′(fn) = e′(
+k
+∑
+n=1
+f(n)f ′(fn))
+since f ∈ FE′(N,E), implying Rt
+f(f ′
+k) ∈ J (E). Hence we can apply Theorem 5.2.15
+(ii) and obtain that RN,E′ is surjective, finishing the proof of part a)(1).
+Thus there is u ∈ F(Ω)εE such that RN,E′(SF(Ω)(u)) = g in both cases. Then
+(T KεidE)(u) ∈ ℓ(N)εE and from (61) we derive
+Sℓ(N)((T KεidE)(u)) = RN,G(SF(Ω)(u)) = g,
+proving the surjectivity of Sℓ(N).
+c) First, we note that the map T E is well-defined. Indeed, we have (e′○T E)(f) =
+T K(e′ ○f) ∈ ℓ(N) for all f ∈ F(Ω,E) and e′ ∈ E′ by the strength of the family. Part
+a) implies that T E(f) ∈ FG(N,E) = ℓ(N,E) and thus the map T E is well-defined
+and its linearity follows from the linearity of the T E
+n for n ∈ N. Next, we prove that
+T E is surjective. Let g ∈ ℓ(N,E). Since T KεidE is an isomorphism and Sℓ(N) by
+part b) as well, we obtain that u ∶= ((T KεidE)−1 ○ S−1
+ℓ(N))(g) ∈ F(Ω)εE. Therefore
+SF(Ω)(u) ∈ F(Ω,E) and from (61) we get
+T E(SF(Ω)(u)) = (T E ○ SF(Ω))(u) = (Sℓ(N) ○ (T KεidE))(u) = g,
+which means that T E is surjective. The injectivity of T E by Proposition 5.2.8,
+implies that
+SF(Ω) = (T E)−1 ○ (Sℓ(N) ○ (T KεidE)),
+yielding the surjectivity of SF(Ω) and thus the ε-compatibility of F(Ω) and F(Ω,E).
+Furthermore, we have T E = Sℓ(N) ○ (T KεidE) ○ S−1
+F(Ω), resulting in T E being an
+isomorphism.
+□
+
+132
+5. APPLICATIONS
+We note that one should not confuse the coefficient space ℓ(N) of the Schauder
+series expansion of functions from F(Ω) in the theorem above with the space
+ℓ1 = ℓ1(N) of absolutely summable sequences.
+We remark again (see Theorem
+5.6.1) that the index set of the equicontinuous Schauder basis of F(Ω) in Theorem
+5.7.1 need not be N (or N0) but may be any other countable index set as long as
+the equicontinuous Schauder basis is unconditional which is, for instance, always
+fulfilled if F(Ω) is nuclear by [89, 21.10.1 Dynin-Mitiagin Theorem, p. 510].
+Theorem 5.7.1 (i) gives another proof of Theorem 5.6.13 b) and Theorem 5.6.14
+b).
+Let us demonstrate an application of the preceding theorem which relates
+the space of O(DR(0),E), 0 < R ≤ ∞, of holomorphic functions on DR(0) with
+values in a complex locally complete lcHs E (see Theorem 5.6.11) and the Köthe
+space λ∞(AR,E) with Köthe matrix AR ∶= (rk
+j )k∈N0,j∈N for some strictly increasing
+sequence (rj)j∈N in (0,R) converging to R (see Corollary 4.2.3), using the sequence
+of Taylor coefficients of a holomorphic function.
+5.7.2. Corollary. Let E be a locally complete lcHs over C, 0 < R ≤ ∞ and
+define the Köthe matrix AR ∶= (rk
+j )k∈N0,j∈N for some strictly increasing sequence
+(rj)j∈N in (0,R) converging to R. Then λ∞(AR)εE ≅ λ∞(AR,E) and
+λE∶O(DR(0),E) → λ∞(AR,E), λE(f) ∶= ((∂k
+C)Ef(0)
+k!
+)
+k∈N0,
+is an isomorphism with λE = Sλ∞(AR) ○ (λCεidE) ○ S−1
+O(DR(0)).
+Proof. By Proposition 4.2.17 and (23) the spaces O(DR(0)) and O(DR(0),E)
+are ε-compatible. Moreover, λ∞(AR) and λ∞(AR,E) are ε-compatible by Corollary
+4.2.3 as limk→∞( rj
+rj+1 )k = 0 for any j ∈ N. Clearly, we have e′ ○ x ∈ λ∞(AR) for all
+e′ ∈ E′ and x ∈ λ∞(AR,E). The space O(DR(0)) with the topology τc of compact
+convergence is a nuclear Fréchet space and thus a Fréchet–Schwartz space.
+In
+particular, this space is barrelled and its Schauder basis of monomials (z ↦ zk)k∈N0
+is equicontinuous. The corresponding coefficient functionals are given by λC
+k and
+the map λC is an isomorphism by [131, Example 27.27, p. 341–342]. By the proof
+of Theorem 5.6.11 the family (λE,λC) is consistent for (O,E) and its strength
+follows from Proposition 5.2.32. Now, we can apply Theorem 5.7.1 (i), yielding our
+statement.
+□
+Let us present another application of Theorem 5.7.1 to the space OM(Rd,E) of
+multipliers for the Schwartz space from Example 3.1.9 d). For simplicity we restrict
+to the case d = 1. Fix a compactly supported test function ϕ ∈ C∞
+c (R) with ϕ(x) = 1
+for x ∈ [0, 1
+4] and ϕ(x) = 0 for x ≥ 1
+2. For f ∈ C∞(R,E) we set
+fj(x) ∶= f(x + j) −
+∞
+∑
+k=0
+akϕ(−2k(x − 1))f(−2k(x − 1 + j) + 1), x ∈ [0,1], j ∈ Z,
+where
+ak ∶=
+∞
+∏
+j=0,j≠k
+1 + 2j
+2j − 2k , k ∈ N0.
+Fixing x ∈ [0,1), we observe that fj(x) is well-defined for each j ∈ Z since there are
+only finitely many summands due to the compact support of ϕ and −2k(x−1) → ∞
+for k → ∞. For x = 1 we have fj(1) = 0 for each j and the convergence of the series
+in E follows from the uniform continuity of f on [0,1], f(0) = 0 and ∑∞
+k=0 ak = 1 by
+the case n = 0 in [160, Lemma (iii), p. 625]. For each e′ ∈ E′ and j ∈ Z we note that
+e′(fj(x)) = (e′ ○f)(x+j)−
+∞
+∑
+k=0
+akϕ(−2k(x−1))(e′ ○f)(−2k(x−1+j)+1), x ∈ [0,1],
+
+5.7. REPRESENTATION BY SEQUENCE SPACES
+133
+which implies that e′ ○fj ∈ E0 by [11, Proposition 3.2, p. 15]. Using the weak-strong
+principle Corollary 5.2.24, we obtain that fj ∈ E0(E) for all j ∈ Z if E is locally
+complete. Setting
+ρ∶R → [0,1], ρ(x) ∶= 1 − cos(arctan(x)) = 1 −
+1
+√
+1 + x2 ,
+we deduce from the proof and with the notation of [12, Proposition 2.2, p. 1494]
+that e′○fj ○ρ = (Φ−1
+2 ○Φ1)(e′○fj) is an element of the Schwartz space S(R) for each
+e′ ∈ E′. The weak-strong principle Corollary 5.2.21 c) yields that fj ○ ρ ∈ S(R,E)
+if E is locally complete. Hence (fj ○ ρ) ⋅ h2n is Pettis-integrable on R for every
+j ∈ Z and n ∈ N0 by Proposition 5.6.12 if E is locally complete where hn is the n-th
+Hermite function. Therefore the Pettis-integral
+bn,j(f) ∶= ⟨fj ○ ρ,h2n⟩L2 ∶= ∫
+R
+fj(ρ(x))h2n(x)dx, j ∈ Z, n ∈ N0,
+is a well-defined element of E by Proposition 5.6.12 if E is locally complete. By
+[12, Theorem 2.1, p. 1496–1497] (cf. [172, Theorem 3, p. 478]) the map
+ΦK∶OM(R) → s(N)′
+b̂⊗πs(N), ΦK(f) ∶= (bσ(n,j)(f))(n,j)∈N2,
+is an isomorphism where σ∶N2 → N0 × Z is the enumeration given by σ(n,j) ∶=
+(n − 1,(j − 1)/2) if j is odd, and σ(n,j) ∶= (n − 1,−j/2) if j is even. Here, we have
+to interpret ΦK(f) as an element of s(N)′
+b̂⊗πs(N) by identification of isomorphic
+spaces. Namely,
+s(N)′
+b̂⊗πs(N) ≅ s(N)̂⊗πs(N)′
+b ≅ s(N)εs(N)′
+b ≅ s(N,s(N)′
+b)
+holds where the first isomorphism is due to the commutativity of ̂⊗π, the second
+due to the nuclearity of s(N) and the last due to Corollary 4.2.3 b) via Ss(N). Then
+we interpret ΦK(f) as an element of s(N,s(N)′
+b) by means of
+j ∈ N �→ [a ∈ s(N) ↦ ∑
+n∈N
+anbσ(n,j)]
+(see also (62) below).
+5.7.3. Corollary. If E is a sequentially complete lcHs, then the map
+ΦE∶OM(R,E) → s(N,Lb(s(N),E)), ΦE(f) ∶= (bσ(n,j)(f))(n,j)∈N2,
+is an isomorphism where we interpret ΦE(f) as an element of s(N,Lb(s(N),E)).
+Proof. The spaces OM(R) and OM(R,E) are ε-compatible by Corollary
+3.2.10 with the inverse of SOM(R) given by the map Rt∶OM(R,E) → OM(R)εE,
+f ↦ J −1 ○ Rt
+f, according to Theorem 3.2.4. The barrelled nuclear space OM(R)
+has the equicontinuous unconditional Schauder basis (ψσ(n,j))(n,j)∈N2 with asso-
+ciated coefficient functionals δn,j ○ ΦK = bσ(n,j) given in [12, Proposition 3.2, p.
+1499]. Next, we show that (ΦE,ΦK) is a strong, consistent family for (OM,E). Let
+f ∈ OM(R,E). For each e′ ∈ E′ and (n,j) ∈ N2 we have
+δn,j ○ ΦK(e′ ○ f) = bσ(n,j)(e′ ○ f) = ∫
+R
+(e′ ○ f)(j−1)/2(ρ(x))h2(n−1)(x)dx
+= ⟨e′,∫
+R
+f(j−1)/2(ρ(x))h2(n−1)(x)dx⟩ = ⟨e′,δn,j ○ ΦE(f)⟩
+= e′(bσ(n,j)(f))
+if j is odd since (f(j−1)/2 ○ ρ) ⋅ h2(n−1) is Pettis-integrable on R. The analogous
+result holds for even j as well. This implies the strength of the family. Due to
+
+134
+5. APPLICATIONS
+Proposition 4.3.3 with (T E
+0 ,T K
+0 ) given by T E
+0 (f) ∶= (fj ○ ρ)h2n, f ∈ OM(R,E), and
+T K
+0 (f) ∶= (fj ○ ρ)h2n, f ∈ OM(R), the family (ΦE,ΦK) is consistent.
+In order to apply Theorem 5.7.1 we need spaces ℓV(N2) and ℓV(N2,E) of
+sequences with values in K and E, respectively.
+In addition, the space ℓV(N2)
+has to be isomorphic to s(N,s(N)′
+b) so that ΦK∶OM(R) → s(N,s(N)′
+b) ≅ ℓV(N2)
+becomes the isomorphism we need for Theorem 5.7.1. We set
+ℓV(N2,E) ∶= {x = (xn,j) ∈ EN2 ∣ ∀ k ∈ N, B ⊂ s(N) bounded, α ∈ A ∶ ∥x∥k,B,α < ∞}
+where
+∥x∥k,B,α ∶=
+sup
+(j,a)∈ωB
+pα(T E(x)(j,a))νk,B(j,a)
+with ωB ∶= N × B and νk,B∶ωB → [0,∞), νk,B(j,a) ∶= (1 + j2)k/2, and
+T E(x)(j,a) ∶= ∑
+n∈N
+anxn,j.
+We claim that the map
+T E∶ℓV(N2,E) → s(N,Lb(s(N),E)), x ↦ (T E(x)(j,⋅))j∈N,
+(62)
+is an isomorphism. We remark for each k ∈ N, bounded B ⊂ s(N) and α ∈ A that
+∣T E(x)∣s(N),k,(B,α) = sup
+j∈N
+sup
+a∈B
+pα(T E(x)(j,a))(1 + j2)k/2 = ∥x∥k,B,α
+for all x ∈ ℓV(N2,E), implying that T E is an isomorphism into. Let y ∶= (yj) ∈
+s(N,Lb(s(N),E)). Then yj ∈ Lb(s(N),E) for j ∈ N and we set xn,j ∶= yj(en) for n ∈
+N where en is the n-th unit sequence in s(N). We note that with x ∶= (xn,j)(n,j)∈N2
+T E(x)(j,a) = ∑
+n∈N
+anxn,j = ∑
+n∈N
+anyj(en) = yj(∑
+n∈N
+anen) = yj(a)
+holds for all j ∈ N and a ∶= (an) ∈ s(N) since (en) is a Schauder basis of s(N) with
+associated coefficient functionals a ↦ an. It follows that x ∈ ℓV(N2,E) and the
+surjectivity of T E.
+The next step is to prove that ℓV(N2) and ℓV(N2,E) are ε-into-compatible. Due
+to Theorem 3.1.12 we only need to show that (T E,T K) is a consistent generator
+for (ℓV,E). Let u ∈ ℓV(N2)εE. Then
+m
+∑
+n=1
+anSℓV(N2)(u)(j,n) =
+m
+∑
+n=1
+anu(δj,n) = u(
+m
+∑
+n=1
+anδj,n)
+(63)
+for all m ∈ N and a ∶= (an) ∈ s(N). Since
+(
+m
+∑
+n=1
+anδj,n)(x) =
+m
+∑
+n=1
+anxj,n → T K(x)(j,a) = T K
+(j,a)(x),
+m → ∞,
+for all x ∈ ℓV(N2), we deduce that (∑m
+n=1 anδj,n)m converges to T K
+(j,a)(x) in ℓV(N2)′
+κ
+by the Banach–Steinhaus theorem, which is applicable as ℓV(N2) ≅ s(N,s(N)′
+b) ≅
+OM(R) is barrelled. We conclude that
+u(T K
+(j,a)) = lim
+m→∞u(
+m
+∑
+n=1
+anδj,n) =
+(63)
+∞
+∑
+n=1
+anSℓV(N2)(u)(j,n) = T ESℓV(N2)(u)(j,a)
+and thus the consistency of (T E,T K) for (ℓV,E).
+Furthermore, we clearly have e′ ○ x ∈ ℓV(N2) for all x ∈ ℓV(N2,E) and the map
+Φ∶OM(R) → s(N)′
+b̂⊗πs(N) ≅ ℓV(N2) is an isomorphism by [12, Theorem 2.1, p.
+1496–1497] and (62). Due to [83, Chap. II, §4, n○4, Théorème 16, p. 131] the dual
+OM(R)′
+b is an LF-space and thus OM(R) ≅ (OM(R)′
+b)′
+b is the strong dual of an
+LF-space by reflexivity and therefore webbed by [94, Satz 7.25, p. 165]. Finally, we
+can apply Theorem 5.7.1 (ii), yielding our statement.
+□
+
+5.7. REPRESENTATION BY SEQUENCE SPACES
+135
+5.7.4. Remark. The actual isomorphism in Corollary 5.7.3 (without the inter-
+pretation) is given by ̃ΦE ∶= T E ○ ΦE with T E from (62) and we have
+̃ΦE = T E ○ ΦE = T E ○ SℓV(N2) ○ (ΦKεidE) ○ S−1
+OM(R).
+Furthermore, Corollary 5.7.3 is valid for locally complete E as well. Indeed, similar
+to Example 4.2.2 we may show that ℓV(N2,E) ≅ ℓV(N2)εE for locally complete E.
+In combination with Corollary 3.2.10 and Theorem 5.1.2 b) this proves Corollary
+5.7.3 for locally complete E as in Theorem 5.6.13 b).
+
+
+Appendices
+137
+
+
+APPENDIX A
+Compactness of closed absolutely convex hulls and
+Pettis-integrals
+A.1. Compactness of closed absolutely convex hulls
+In this section of the appendix we treat the question for which functions f∶Ω →
+E, subsets K ⊂ Ω and lcHs E sets like acx(f(K)) are compact or sets like
+Nj,m(f) ∶= {T E
+m(f)(x)νj,m(x) ∣ x ∈ ωm},
+j ∈ J, m ∈ M,
+for f ∈ FV(Ω,E) are contained in an absolutely convex compact set. This is useful
+in connection with ε-compatibility due to Corollary 3.2.5 (iv) and also relevant
+in connection with the Pettis-integrability of a vector-valued function due to the
+Mackey–Arens theorem.
+We recall that the space of càdlàg functions on a set Ω ⊂ R with values in an
+lcHs E is defined by
+D(Ω,E) ∶= {f ∈ EΩ ∣ ∀ x ∈ Ω ∶
+lim
+w→x+f(w) = f(x) and f(x−) ∶= lim
+w→x−f(w) exists}.4
+A.1.1. Proposition. Let Ω ⊂ R, K ⊂ Ω be compact and E an lcHs. Then f(K)
+is precompact for every f ∈ D(Ω,E). If E is quasi-complete, then acx(f(K)) is
+compact.
+Proof. Let f ∈ D(Ω,E), α ∈ A and ε > 0. We recall and define
+Br(x) = {w ∈ R ∣ ∣w − x∣ < r}
+and
+Bε,α(y) ∶= {w ∈ E ∣ pα(w − y) < ε}
+for every x ∈ Ω, y ∈ E and r > 0. Let x ∈ Ω. Then there is rx− > 0 such that
+pα(f(w) − f(x−)) < ε for all w ∈ Brx−(x) ∩ (−∞,x) ∩ Ω if x is an accumulation
+point of (−∞,x] ∩ Ω. Further, there is rx+ > 0 such that pα(f(w) − f(x)) < ε for
+all w ∈ Brx+(x) ∩ [x,∞) ∩ Ω if x is an accumulation point of [x,∞) ∩ Ω. If x is an
+accumulation point of (−∞,x]∩Ω and [x,∞)∩Ω, we choose rx ∶= min(rx−,rx+). If x
+is an accumulation point of (−∞,x]∩Ω but not of [x,∞)∩Ω, we choose rx ∶= rx−. If
+x is an accumulation point of [x,∞)∩Ω but not of (−∞,x]∩Ω, we choose rx ∶= rx+.
+If x is neither an accumulation point of (−∞,x] ∩ Ω nor of [x,∞) ∩ Ω, then there is
+rx > 0 such that Brx(x) ∩ Ω = {x}.
+Setting Vx ∶= Brx(x)∩Ω, we note that the sets Vx are open in Ω with respect to
+the topology induced by R and K ⊂ ⋃x∈K Vx. Since K is compact, there are n ∈ N
+and x1,...,xn ∈ K such that K ⊂ ⋃n
+i=1 Vxi. W.l.o.g. each xi is an accumulation point
+of (−∞,xi]∩Ω and [xi,∞)∩Ω. Then we have f(w) ∈ (Bε,α(f(xi−))∪Bε,α(f(xi)))
+for all w ∈ Vxi and get
+f(K) ⊂
+n
+⋃
+i=1
+f(Vxi) ⊂
+n
+⋃
+i=1
+(Bε,α(f(xi−)) ∪ Bε,α(f(xi))),
+which means that f(K) is precompact.
+4We recall that for x ∈ Ω we only demand limw→x+ f(w) = f(x) if x is an accumulation
+point of [x, ∞) ∩ Ω, and the existence of the limit limw→x− f(w) if x is an accumulation point of
+(−∞, x] ∩ Ω.
+139
+
+140
+A. COMPACTNESS OF CLOSED ABS. CONVEX HULLS & PETTIS-INTEGRALS
+If E is quasi-complete, then the precompact set f(K) is relatively compact
+by [89, 3.5.3 Proposition, p. 65]. Hence acx(f(K)) is compact as quasi-complete
+spaces have ccp.
+□
+For f ∈ D(Ω,E) we define the jump function ∆∗f(x) ∶= f(x) − f(x−), x ∈ Ω,
+where we set f(x−) ∶= 0 if x is not an accumulation point of (−∞,x] ∩ Ω.
+A.1.2. Proposition. Let Ω ⊂ R, K ⊂ Ω be compact and E an lcHs. Then
+∆∗f(K) is precompact for every f ∈ D(Ω,E). If E is quasi-complete, then the set
+acx(∆∗f(K)) is compact.
+Proof. If K is a finite set, then ∆∗f(K) is finite, thus compact, and we are
+done. So let us assume that K is not finite. Let α ∈ A and ε > 0. We define
+∆ε,α ∶= {x ∈ K ∣ pα(∆∗f(x)) ≥ ε} and claim that ∆α,ε is a finite set.
+Let us
+assume the contrary. Then there is an infinite sequence (xn) in ∆ε,α ⊂ K. Due
+to the compactness of K there is a subsequence of (xn) which converges to some
+x ∈ K. W.l.o.g. this subsequence is strictly increasing and we call this subsequence
+again (xn). Since f has left limits (in left-accumulation points), for every n ∈ N,
+n ≥ 2, there is wn ∈ (xn−1,xn) such that pα(f(xn−) − f(wn)) ≤ ε/2 (if xn is not an
+accumulation point of (−∞,xn] ∩ Ω, then there is wn ∈ (xn−1,xn) with wn ∉ Ω and
+we set f(wn) ∶= 0). Hence we have
+pα(f(xn) − f(wn)) ≥ pα(f(xn) − f(xn−)) − pα(f(xn−) − f(wn))
+= pα(∆∗f(xn)) − pα(f(xn−) − f(wn)) ≥ ε/2
+for all n ≥ 2. But this is a contradiction because
+lim
+n→∞f(xn) = lim
+n→∞f(wn) = f(x−),
+which proves our claim.
+Next, we note that
+∆∗f(K) ⊂ (Bε,α(0) ∪ ∆∗f(∆ε,α)) ⊂
+⋃
+z∈{0}∪∆∗f(∆ε,α)
+z + Bε,α(0),
+which implies that ∆∗f(K) is precompact as {0} ∪ ∆∗f(∆ε,α) is finite.
+If E is quasi-complete, then the precompact set ∆∗f(K) is relatively compact
+by [89, 3.5.3 Proposition, p. 65]. Hence acx(∆∗f(K)) is compact as quasi-complete
+spaces have ccp.
+□
+Proposition A.1.1 and Proposition A.1.2 are known in the case that Ω = [0,1]
+and E = K (see the comments after [19, Chap. 3, Sect. 14, Lemma 1, p. 110]) since
+precompactness is equivalent to boundedness if E = K.
+A.1.3. Proposition. Let Ω be a locally compact topological Hausdorff space
+and f ∈ C0(Ω,E). If
+(i) E is an lcHs with ccp, or
+(ii) E is an lcHs with metric ccp and Ω second-countable,
+then acx(f(Ω)) is compact.
+Proof. Let Ω be compact, then f(Ω) is compact in E as f is continuous. If
+Ω is even second-countable, then Ω is metrisable by [58, Chap. XI, 4.1 Theorem, p.
+233] and thus f(Ω) as well by [34, Chap. IX, §2.10, Proposition 17, p. 159]. This
+yields that acx(f(Ω)) is compact in both cases.
+Let Ω be non-compact and Ω∗ denote the one-point compactification of Ω.
+Since f ∈ C0(Ω,E), it has a unique continuous extension ̂f to Ω∗ with ̂f(∞) = 0.
+Hence K ∶= ̂f(Ω∗) is a compact set in E as Ω∗ is compact and ̂f continuous. If
+Ω is even second-countable, then Ω∗ is metrisable by [58, Chap. XI, 8.6 Theorem,
+
+A.1. COMPACTNESS OF CLOSED ABSOLUTELY CONVEX HULLS
+141
+p. 247] and thus K as well by [34, Chap. IX, §2.10, Proposition 17, p. 159]. This
+yields that acx(K) is compact in both cases and thus the closed subset acx(f(Ω)),
+too.
+□
+We note that C0(Ω,E) = C(Ω,E) if Ω is compact. For our next proposition we
+define the space of bounded γ-Hölder continuous functions, 0 < γ ≤ 1, from a metric
+space (Ω,d) to an lcHs E by
+C[γ]
+b
+(Ω,E) ∶= {f ∈ EΩ ∣ ∀α ∈ A ∶ sup
+x∈Ω
+pα(f(x)) < ∞ and sup
+x,y∈Ω
+x≠y
+pα(f(x) − f(y))
+d(x,y)γ
+< ∞}.
+A.1.4. Proposition. Let (Ω,d) be a metric space, E a locally complete lcHs
+and f ∈ C[γ]
+b
+(Ω,E) for some 0 < γ ≤ 1. If there is h∶Ω → (0,∞) such that fh is
+bounded on Ω and with N ∶= {x ∈ Ω ∣ f(x) = 0} it holds that
+∀ ε > 0 ∃ K ⊂ Ω compact ∀ x ∈ Ω ∖ (K ∪ N) ∶ 1 ≤ εh(x),
+then acx(f(Ω)) is compact.
+Proof. Since f ∈ C[γ]
+b
+(Ω,E), the sets f(Ω) and
+B1 ∶= {f(z) − f(t)
+d(z,t)γ
+∣ z,t ∈ Ω, z ≠ t}
+are bounded in E. Further, the range (fh)(Ω) is bounded in E by assumption.
+Thus B ∶= acx(B1 ∪ f(Ω) ∪ (fh)(Ω)) is a closed disk and EB a Banach space with
+the norm ∥x∥B ∶= inf{r > 0 ∣ x ∈ rB}, x ∈ EB, as E is locally complete. Next,
+we show that f(Ω) is precompact in EB. Let V be a zero neighbourhood in EB.
+Then there is ε > 0 such that Uε ∶= {x ∈ EB ∣ ∥x∥B ≤ ε} ⊂ V . Moreover, there is a
+compact set K ⊂ Ω such that 1 ≤ εh(x) for all x ∈ Ω∖(K ∪N). The map f∶Ω → EB
+is well-defined and uniformly continuous because ∥f(z) − f(t)∥B ≤ d(z,t)γ for all
+z,t ∈ Ω, which follows from B1 ⊂ B. We deduce that f(K) is compact in EB. We
+note that
+f(x) = f(x)h(x)
+1
+h(x),
+x ∈ Ω ∖ N,
+which implies that ∥f(x)∥B ≤
+1
+h(x) as (fh)(Ω) ⊂ B. Hence we have
+∥f(x)∥B ≤
+1
+h(x) ≤ ε,
+x ∈ Ω ∖ (K ∪ N),
+and the estimate 0 = ∥f(x)∥B ≤ ε is still valid for x ∈ N, yielding f(Ω ∖ K) ⊂ Uε.
+Since f(K) is compact in EB, it is also precompact and so there is a finite set
+P ⊂ EB such that f(K) ⊂ P + V . We derive that
+f(Ω) = (f(K) ∪ f(Ω ∖ K)) ⊂ ((P + V ) ∪ Uε) ⊂ ((P ∪ {0}) + V ),
+which means that f(Ω) is precompact in EB and thus acx(f(Ω)) as well by [89,
+6.7.1 Proposition, p. 112]. Therefore the set acx(f(Ω)) is compact in the Banach
+space EB and also compact in the weaker topology of E.
+□
+The underlying idea of Proposition A.1.4 is taken from [29, Lemma 1, Propo-
+sition 2, p. 354].
+A.1.5. Proposition. Let Ω ⊂ Rd be an open convex set, E an lcHs over K and
+f∶Ω → E weakly C1
+b , i.e. e′ ○ f ∈ C1
+b (Ω) for each e′ ∈ E′. Then f ∈ C[1]
+b (Ω,E).
+Proof. Let z,t ∈ Ω, z ≠ t. By the mean value theorem we have
+∣(e′ ○ f)(z) − (e′ ○ f)(t)∣
+∣z − t∣
+≤ Cd max
+1≤n≤dsup
+x∈Ω
+∣(∂en)K(e′ ○ f)(x)∣ ≤ Cd∣e′ ○ f∣C1
+b (Ω) < ∞
+
+142
+A. COMPACTNESS OF CLOSED ABS. CONVEX HULLS & PETTIS-INTEGRALS
+for all e′ ∈ E′ where Cd ∶=
+√
+d if K = R, and Cd ∶= 2
+√
+d if K = C. It follows from
+[131, Mackey’s theorem 23.15, p. 268] that f is Lipschitz continuous and bounded
+as well, thus f ∈ C[1]
+b (Ω,E).
+□
+A.1.6. Proposition. Let FV(Ω,E) be a dom-space, let there be a set X, a
+family K of sets and a map π∶⋃m∈M ωm → X such that ⋃K∈K K ⊂ X.
+If f ∈
+FV(Ω,E) fulfils
+∀ ε > 0, j ∈ J, m ∈ M, α ∈ A ∃ K ∈ K ∶
+(i)
+sup
+x∈ωm,
+π(x)∉K
+pα(T E
+m(f)(x))νj,m(x) < ε,
+(ii) Nπ⊂K,j,m(f) ∶= {T E
+m(f)(x)νj,m(x) ∣ x ∈ ωm, π(x) ∈ K} is precompact in E,
+then the set Nj,m(f) is precompact in E for every j ∈ J and m ∈ M.
+If E is
+quasi-complete, then acx(Nj,m(f)) is compact.
+Proof. Let V be a zero neighbourhood in E. Then there are α ∈ A and ε > 0
+such that Bε,α ⊂ V where Bε,α ∶= {x ∈ E ∣ pα(x) < ε}. Let j ∈ J and m ∈ M. Due to
+(i) there is K ∈ K such that the set
+Nπ⊄K,j,m(f) ∶= {T E
+m(f)(x)νj,m(x) ∣ x ∈ ωm, π(x) ∉ K}
+is contained in Bε,α. Further, the precompactness of Nπ⊂K,j,m(f) by (ii) implies
+that there exists a finite set P ⊂ E such that Nπ⊂K,j,m(f) ⊂ P + V . Hence we
+conclude
+Nj,m(f) = (Nπ⊄K,j,m(f) ∪ Nπ⊂K,j,m(f))
+⊂ (Bε,α ∪ (P + V )) ⊂ (V ∪ (P + V )) = (P ∪ {0}) + V,
+which means that Nj,m(f) is precompact.
+The second part of the statement follows from the fact that a precompact set
+in a quasi-complete space is relatively compact by [89, 3.5.3 Proposition, p. 65] and
+that quasi-complete spaces have ccp.
+□
+The most common case is that K consists of the compact subsets of Ω and π
+is a projection on X ∶= Ω (see e.g. Example 4.2.11, Example 4.2.16 and Example
+4.2.22).
+A.2. The Pettis-integral
+We start with the definition of the Pettis-integral which we use to define Fourier
+transformations of vector-valued functions (see Proposition 4.2.25, Theorem 5.6.13
+and Theorem 5.6.14) and for Riesz–Markov–Kakutani theorems in Section 4.3.
+Let Σ be a σ-algebra on a set X.
+A function µ∶Σ → K is called K-valued
+measure if µ(∅) = 0 and µ is countably additive, i.e. for any sequence (An)n∈N of
+pairwise disjoint sets in Σ it holds that
+µ( ⋃
+n∈N
+An) = ∑
+n∈N
+µ(An) ∈ K.
+If K = R, µ is also called a signed measure, and if K = C a complex measure. If K is
+replaced by [0,∞], we say that µ is a positive measure. For a K-valued measure µ
+its total variation ∣µ∣ given by
+∣µ∣(A) ∶= sup{∑
+n∈N
+∣µ(An)∣ ∣ An ∈ Σ,Am ∩ An = ∅ if m ≠ n,A = ⋃
+n∈N
+An},
+A ∈ Σ,
+
+A.2. THE PETTIS-INTEGRAL
+143
+is a well-defined positive measure by [149, 6.2 Theorem, p. 117] and it is finite by
+[149, 6.4 Theorem, p. 118], i.e. ∣µ∣(X) < ∞. Obviously, a K-valued measure µ is
+positive if and only if ∣µ∣ = µ. For a positive measure µ on X and 1 ≤ p < ∞ let
+Lp(X,µ) ∶= {f∶X → K measurable ∣ qp(f) ∶= ∫
+X
+∣f(x)∣pdµ(x) < ∞}
+and define the quotient space of p-integrable functions by Lp(X,µ) ∶= Lp(X,µ)/{f ∈
+Lp(X,µ) ∣ qp(f) = 0}, which becomes a Banach space if it is equipped with the norm
+∥f∥p ∶= ∥f∥Lp ∶= qp(F)1/p, f = [F] ∈ Lp(X,µ). From now on we do not distinguish
+between equivalence classes and their representatives anymore.
+For a K-valued measure µ there is a unique h ∈ L1(X,∣µ∣) with dµ = hd∣µ∣ by
+the Radon–Nikodým theorem (see [149, 6.12 Theorem, p. 124]) and h can be chosen
+such that ∣h∣ = 1, i.e. has a representative with modulus equal to 1. Now, we say
+that f ∈ Lp(X,µ) if f ⋅ h ∈ Lp(X,∣µ∣). For f ∈ L1(X,µ) we define the integral of f
+on X w.r.t. µ by
+∫
+X
+f(x)dµ(x) ∶= ∫
+X
+f(x)h(x)d∣µ∣(x).
+For a measure space (X,Σ,µ) and f∶X → K we say that f is integrable on
+Λ ∈ Σ and write f ∈ L1(Λ,µ) if χΛf ∈ L1(X,µ). Then we set
+∫
+Λ
+f(x)dµ(x) ∶= ∫
+X
+χΛ(x)f(x)dµ(x).
+A.2.1. Definition (Pettis-integral). Let (X,Σ,µ) be a measure space and E
+an lcHs. A function f∶X → E is called weakly measurable if the function e′ ○f∶X →
+K, (e′ ○ f)(x) ∶= ⟨e′,f(x)⟩ ∶= e′(f(x)), is measurable for all e′ ∈ E′.
+A weakly
+measurable function is said to be weakly integrable if e′ ○ f ∈ L1(X,µ). A function
+f∶X → E is called Pettis-integrable on Λ ∈ Σ if it is weakly integrable on Λ and
+∃ eΛ ∈ E ∀e′ ∈ E′ ∶ ⟨e′,eΛ⟩ = ∫
+Λ
+⟨e′,f(x)⟩dµ(x).
+In this case eΛ is unique due to E being Hausdorff and we set the Pettis-integral
+∫
+Λ
+f(x)dµ(x) ∶= eΛ.
+If we consider the measure space (X,L (X),λ) of Lebesgue measurable sets
+for X ⊂ Rd, we just write dx ∶= dλ(x).
+A.2.2. Lemma. Let E be a locally complete lcHs, Ω ⊂ Rd open and f∶Ω → E.
+If f is weakly C1, i.e. e′ ○ f ∈ C1(Ω) for every e′ ∈ E′, then f is Pettis-integrable on
+every compact subset K ⊂ Ω with respect to any locally finite positive measure µ on
+Ω and
+pα(∫
+K
+f(x)dµ(x)) ≤ µ(K)sup
+x∈K
+pα(f(x)),
+α ∈ A.
+Proof. Let K ⊂ Ω be compact and (Ω,Σ,µ) a measure space with locally
+finite measure µ, i.e. Σ contains the Borel σ-algebra B(Ω) on Ω and for every
+x ∈ Ω there is a neighbourhood Ux ⊂ Ω of x such that µ(Ux) < ∞. Since the map
+e′ ○ f is differentiable for every e′ ∈ E′, thus Borel-measurable, and B(Ω) ⊂ Σ, it is
+measurable. We deduce that e′ ○f ∈ L1(K,µ) for every e′ ∈ E′ because locally finite
+measures are finite on compact sets. Therefore the map
+I∶E′ → K, I(e′) ∶= ∫
+K
+⟨e′,f(x)⟩dµ(x)
+
+144
+A. COMPACTNESS OF CLOSED ABS. CONVEX HULLS & PETTIS-INTEGRALS
+is well-defined and linear. We estimate
+∣I(e′)∣ ≤ ∣µ(K)∣ sup
+x∈f(K)
+∣e′(x)∣ ≤ µ(K)
+sup
+x∈acx(f(K))
+∣e′(x)∣,
+e′ ∈ E′.
+Due to f being weakly C1 and [29, Proposition 2, p. 354] the absolutely convex set
+acx(f(K)) is compact, yielding I ∈ (E′
+κ)′ ≅ E by the theorem of Mackey–Arens,
+which means that there is eK ∈ E such that
+⟨e′,eK⟩ = I(e′) = ∫
+K
+⟨e′,f(x)⟩dµ(x),
+e′ ∈ E′.
+Hence f is Pettis-integrable on K w.r.t. µ. For α ∈ A we set Bα ∶= {x ∈ E ∣ pα(x) < 1}
+and observe that
+pα(∫
+K
+f(x)dµ(x)) = sup
+e′∈B○α
+∣⟨e′,∫
+K
+f(x)dµ(x)⟩∣ = sup
+e′∈B○α
+∣∫
+K
+e′(f(x))dµ(x)∣
+≤ µ(K) sup
+e′∈B○α
+sup
+x∈K
+∣e′(f(x))∣ = µ(K)sup
+x∈K
+pα(f(x))
+where we used [131, Proposition 22.14, p. 256] in the first and last equation to get
+from pα to supe′∈B○α and back.
+□
+A.2.3. Lemma. Let E be a sequentially complete lcHs, Ω ⊂ Rd open, (Ω,Σ,µ)
+a measure space with locally finite positive measure µ and f∶Ω → E. If f is weakly
+C1 and there are ψ ∈ L1(Ω,µ) and g∶Ω → [0,∞) measurable such that ψg ≥ 1 and
+fg is bounded on Ω, then f is Pettis-integrable on Ω and
+pα(∫
+Ω
+f(x)dµ(x)) ≤ ∥ψ∥1 sup
+x∈Ω
+pα(f(x)g(x)),
+α ∈ A.
+Proof. Let (Kn)n∈N be a compact exhaustion of Ω. Due to Lemma A.2.2 the
+Pettis-integral
+en ∶= ∫
+Kn
+f(x)dµ(x)
+is a well-defined element of E for every n ∈ N.
+Next, we show that (en) is a
+Cauchy sequence in E.
+Let α ∈ A, m ∈ N0 and k,n ∈ N with k > n.
+We set
+Bα ∶= {x ∈ E ∣ pα(x) < 1} and Qk,n ∶= Kk ∖ Kn and note that
+pα(ek − en) = sup
+e′∈B○α
+∣e′(ek − en)∣ = sup
+e′∈B○α
+∣ ∫
+Qk,n
+e′(f(x))dµ(x)∣
+≤ ∫
+Qk,n
+∣ψ(x)∣dµ(x) sup
+e′∈B○α
+sup
+x∈Ω
+∣e′(f(x)g(x))∣
+= ∫
+Qk,n
+∣ψ(x)∣dµ(x)sup
+x∈Ω
+pα(f(x)g(x))
+(64)
+where we used [131, Proposition 22.14, p. 256] to switch from pα to supe′∈B○α and
+back. Since ψ ∈ L1(Ω,µ), we have that (en) is a Cauchy sequence in the sequentially
+complete space E. Thus eΩ ∶= limn→∞ en exists in E and the dominated convergence
+theorem implies
+e′(eΩ) = lim
+n→∞e′(en) = lim
+n→∞∫
+Kn
+e′(f(x))dµ(x) = ∫
+Ω
+e′(f(x))dµ(x),
+e′ ∈ E′.
+Hence f is Pettis-integrable on Ω with ∫Ω f(x)dµ(x) = eΩ. As in (64) we have
+pα(en) ≤ ∫
+Kn
+∣ψ(x)∣dµ(x)sup
+x∈Ω
+pα(f(x)g(x)) ≤ ∥ψ∥1 sup
+x∈Ω
+pα(f(x)g(x))
+
+A.2. THE PETTIS-INTEGRAL
+145
+for every n ∈ N. Letting n → ∞, we derive the estimate in our statement.
+□
+A.2.4. Remark. Let µ be a K-valued measure and Σ contain B(Ω).
+Then
+Lemma A.2.2 is still valid with µ(K) replaced by ∣µ∣(K) due to the definition of
+the integral w.r.t. a K-valued measure and as ∣µ∣(K) ≤ ∣µ∣(Ω) < ∞. Thus Lemma
+A.2.3 holds in this case as well.
+The following definition is analogous to the definition of the Pettis-integral.
+A.2.5. Definition (Pettis-summable). Let I be a non-empty set and E an
+lcHs. A family (fi)i∈I in E is called weakly summable if (⟨e′,fi⟩)i∈I ∈ ℓ1(I,K) for
+all e′ ∈ E′. A family (fi)i∈I in E is called Pettis-summable if it is weakly summable
+and
+∃ eI ∈ E ∀e′ ∈ E′ ∶ ⟨e′,eI⟩ = ∑
+i∈I
+⟨e′,fi⟩.
+In this case eI is unique due to E being Hausdorff and we set
+∑
+i∈I
+fi ∶= eI.
+For the elements f of the space D([0,1],E) of E-valued càdlàg functions on
+[0,1] and their jump functions ∆∗f we have the following result.
+A.2.6. Proposition. Let E be a quasi-complete lcHs, µ a K-valued Borel mea-
+sure on [0,1] and ψ ∈ ℓ1([0,1],K). Then f ∈ D([0,1],E) is Pettis-integrable on
+[0,1] and
+pα( ∫
+[0,1]
+f(x)dµ(x)) ≤ ∣µ∣([0,1]) sup
+x∈[0,1]
+pα(f(x)),
+α ∈ A,
+and (∆∗f)ψ is Pettis-summable on [0,1] and
+pα( ∑
+x∈[0,1]
+(∆∗f)(x)ψ(x)) ≤ ∥ψ∥ℓ1 sup
+x∈[0,1]
+pα(∆∗f(x)),
+α ∈ A.
+Proof. By [19, Chap. 3, Sect. 14, Lemma 1, p. 110] e′ ○ f is Borel measurable
+for every e′ ∈ E′ and integrable due to its boundedness on [0,1]. Thus the map
+I∶E′ → K, I(e′) ∶= ∫
+[0,1]
+e′(f(x))dµ(x),
+is well-defined and linear. It follows from Proposition A.1.1 that acx(f([0,1])) is
+absolutely convex and compact in E. In combination with the estimate
+∣I(e′)∣ ≤ ∣µ∣([0,1])
+sup
+x∈f([0,1])
+∣e′(x)∣ ≤ ∣µ∣([0,1])
+sup
+x∈acx(f([0,1]))
+∣e′(x)∣
+for every e′ ∈ E′ we deduce that I ∈ (E′
+κ)′ ≅ E by the theorem of Mackey–Arens,
+which implies that there is e[0,1] ∈ E such that
+⟨e′,e[0,1]⟩ = I(e′) = ∫
+[0,1]
+e′(f(x))dµ(x),
+e′ ∈ E′.
+Thus f is Pettis-integrable on [0,1].
+Since ψ ∈ ℓ1([0,1]) and e′ ○ ∆∗f bounded on [0,1] for every e′ ∈ E′, the map
+I0∶E′ → K, I0(e′) ∶=
+∑
+x∈[0,1]
+e′((∆∗f)(x)ψ(x)),
+is well-defined and linear. Moreover, the set acx(∆∗f([0,1])) is absolutely convex
+and compact by Proposition A.1.2. Again, the estimate
+∣I0(e′)∣ ≤
+∑
+x∈[0,1]
+∣ψ(x)∣
+sup
+x∈∆∗f([0,1])
+∣e′(x)∣ ≤ ∥ψ∥ℓ1
+sup
+x∈acx(∆∗f([0,1]))
+∣e′(x)∣
+
+146
+A. COMPACTNESS OF CLOSED ABS. CONVEX HULLS & PETTIS-INTEGRALS
+for every e′ ∈ E′, implies our statement.
+The remaining estimates are deduced
+analogously to Lemma A.2.2.
+□
+A.2.7. Proposition. Let E be an lcHs, Ω a topological Hausdorff space and
+(Ω,Σ,µ) a measure space.
+If f∶Ω → E is weakly integrable and there are ψ ∈
+L1(Ω,µ) and g∶Ω → [0,∞) measurable such that ψg ≥ 1 and
+(i) E has ccp, Ω is locally compact and fg ∈ C0(Ω,E), or
+(ii) E has metric ccp, Ω is locally compact and second-countable, and fg ∈
+C0(Ω,E), or
+(iii) E is locally complete, Ω a metric space, fg ∈ C[γ]
+b
+(Ω,E) for some 0 < γ ≤ 1
+and there is h∶Ω → (0,∞) such that fgh is bounded on Ω and with N ∶=
+{x ∈ Ω ∣ f(x)g(x) = 0} it holds that
+∀ ε > 0 ∃ K ⊂ Ω compact ∀ x ∈ Ω ∖ (K ∪ N) ∶ 1 ≤ εh(x),
+then f is Pettis-integrable on Ω and
+pα(∫
+Ω
+f(x)dµ(x)) ≤ ∥ψ∥1 sup
+x∈Ω
+pα(f(x)g(x)),
+α ∈ A.
+Proof. Since f is weakly integrable, the map
+I∶E′ → K, I(e′) ∶= ∫
+Ω
+e′(f(x))dµ(x),
+is well-defined and linear. It follows from Proposition A.1.3 in case (i)-(ii) and from
+Proposition A.1.4 in case (iii) that acx(fg(Ω)) is absolutely convex and compact
+in E. If µ is a positive measure, i.e. [0,∞]-valued, we observe that
+∣I(e′)∣ ≤ ∫
+Ω
+∣ψ(x)∣dµ(x)
+sup
+x∈fg(Ω)
+∣e′(x)∣ ≤ ∥ψ∥1
+sup
+x∈acx(fg(Ω))
+∣e′(x)∣
+for every e′ ∈ E′. If µ is a K-valued measure, then the same estimate holds with µ
+replaced by ∣µ∣. We deduce from this estimate that I ∈ (E′
+κ)′ ≅ E by the theorem
+of Mackey–Arens, which implies that there is eΩ ∈ E such that
+⟨e′,eΩ⟩ = I(e′) = ∫
+Ω
+e′(f(x))dµ(x),
+e′ ∈ E′.
+Thus f is Pettis-integrable on Ω. The remaining estimate is deduced analogously
+to Lemma A.2.2.
+□
+The idea how to prove Proposition A.2.7 (ii) for Ω = Rd is due to an anonymous
+reviewer of [110] but did not make it into [110] because of page limits.
+
+List of Symbols
+Sets and systems of sets
+acx(M)
+absolutely convex hull of the set M 17
+acx(M)
+closure of the absolutely convex hull of the set M 17
+B○F (Ω)′
+Fν(Ω)
+the polar {y′ ∈ F(Ω)′ ∣ ∀ f ∈ BFν(Ω) ∶ ∣y′(f)∣ ≤ 1} 84
+Br(x)
+ball {w ∈ Rd ∣ ∣w − x∣ < r} around x ∈ Rd with radius r > 0 17
+ch(M)
+circled hull of the set M 17
+cx(M)
+convex hull of the set M 17
+Dr(z)
+disc {w ∈ C ∣ ∣w − z∣ < r} around z ∈ C with radius r > 0 17
+D
+open unit disc D1(0) 19
+G○
+the polar set of G 17
+Mm
+the set {β ∈ Nd
+0 ∣ ∣β∣ ≤ min(m,k)} for m ∈ N0 and k ∈ N∞ 24
+M
+closure of the set M 17
+M
+t
+closure of the set M w.r.t. the topology t 17
+M
+X
+closure of the set M in the topological space X 17
+∂M
+boundary of the set M 17
+N∞
+the set N ∪ {∞} 19
+σ(E,G)
+weak topology induced on E by a separating subspace G ⊂ E′ 78
+τc
+topology of compact convergence 21
+Locally convex Hausdorff spaces & spaces of continuous linear operators
+E′⋆
+algebraic dual of the dual E′ 29
+E
+locally convex Hausdorff space 17
+ED
+space ⋃n∈N nD for a disk D ⊂ E 18
+F ′
+dual space of F 17
+t(F ′,F)
+topology on F ′ where t = b, γ, κ, σ or τ 17
+t(F)
+bornology on F that induces the topology t(F ′,F) 17
+F ≅ E
+locally convex Hausdorff spaces F and E are isomorphic 17
+FεE
+space Le(F ′
+κ,E) where L(F ′
+κ,E) is equipped with the topology of
+uniform convergence on the equicontinuous subsets of F ′ 17
+F ⊗ E
+tensor product of F and E 18
+F ⊗ε E
+F ⊗ E equipped with the topology induced by FεE
+18
+F ̂⊗εE
+completion of F ⊗ε E 18
+F ⊗π E
+F ⊗ E equipped with the projective topology 70
+F ̂⊗πE
+completion of F ⊗π E 70
+L(F,E)
+space of continuous linear operators from F to E 17
+Lt(F,E)
+space L(F,E) equipped with the topology t 17
+b
+topology t = b on L(F,E) of uniform convergence on
+the bounded subsets of F 17
+147
+
+148
+List of Symbols
+γ
+topology t = γ on L(F,E) of uniform convergence on
+the precompact (totally bounded) subsets of F 17
+κ
+topology t = κ on L(F,E) of uniform convergence on
+the absolutely convex, compact subsets of F 17
+σ
+topology t = σ on L(F,E) of uniform convergence on
+the finite subsets of F 17
+τ
+topology t = τ on L(F,E) of uniform convergence on
+the absolutely convex, σ(F,F ′)-compact subsets of F
+17
+(pα)α∈A
+directed system of seminorms inducing the locally convex Hausdorff
+topology on E 17
+Spaces of functions
+Aτ
+∂(C,E)
+space of holomorphic functions f∶C → E of exponential type τ 51
+Aτ
+∆(Rd,E)
+space of harmonic functions f∶Rd → E of exponential type τ 51
+A(Ω,E)
+space of continuous functions f∶Ω → E such that f is holomorphic
+on Ω 47
+Bν(D,E)
+Bloch type space 87
+C(Ω,X)
+space of continuous functions f∶Ω → X 17
+C0(Ω,X)
+space of continuous functions f∶Ω → X that vanish at infinity 17
+c0(A,E)
+space of elements (xk) in the Köthe space λ∞(A,E) such that
+(xkak,j) converges to 0 in E for all j ∈ N 123
+c(N,E)
+space of convergent sequences in E 123
+Cb(Ω,E)
+space of bounded continuous functions f∶Ω → E 46
+C[γ]
+b
+(Ω,E)
+space of bounded γ-Hölder continuous functions f∶Ω → E 140
+C1
+b (Ω,E)
+space of continuously partially differentiable functions f∶Ω → E
+such that (∂β)Ef is bounded on Ω for all ∣β∣ ≤ 1 20
+C∞
+P (∂)(Ω,E)
+kernel of the linear partial differential operator P(∂)E in C∞(Ω,E)
+48
+C∞
+P (∂),b(Ω,E)
+space of bounded functions f in C∞
+P (∂)(Ω,E) 53
+Cu(Ω,E)
+space of uniformly continuous functions f∶Ω → E 39
+Cbu(Ω,E)
+space of bounded uniformly continuous functions f∶Ω → E 45
+CC(Ω,E)
+space of Cauchy continuous functions f∶Ω → E 38
+Cext(Ω,E)
+space of continuous functions f∶Ω → E which have a continuous
+extension to Ω 39
+C[γ]
+z (Ω,E)
+space of γ-Hölder continuous functions f∶Ω → E such that f(z) = 0
+45
+C[γ]
+z,0(Ω,E)
+space of functions f in C[γ]
+z (Ω,E) that vanish at infinity 45
+Ck(Ω,E)
+space of k-times continuously partially differentiable functions
+f∶Ω → E 19
+C∞
+2π(Rd,E)
+space of functions f in C∞(Rd,E) which are 2π-periodic in each
+variable 58
+Ck(Ω,E)
+space of functions f in Ck(Ω,E) such that all partial derivatives
+(∂β)Ef up to order k are continuously extendable on Ω 58
+Ck,γ(Ω,E)
+space of functions f in Ck(Ω,E) such that all partial derivatives
+(∂β)Ef of order k are γ-Hölder continuous 105
+Ck,γ
+loc (Ω,E)
+space of functions f in Ck(Ω,E) such that all partial derivatives
+(∂β)Ef of order k are locally γ-Hölder continuous 107
+CVk(Ω,E)
+space of functions f in Ck(Ω,E) s.t. (β,x) ↦ (∂β)Ef(x)νj,m(β,x)
+is bounded on ωm for all j ∈ J and m ∈ N0 24
+
+List of Symbols
+149
+CWk(Ω,E)
+space Ck(Ω,E) equipped with the topology of uniform convergence
+of partial derivatives up to order k on compact subsets of Ω 25
+CVk
+P (∂)(Ω,E)
+kernel of the linear partial differential operator P(∂)E in CVk(Ω,E)
+26
+CVk
+0(Ω,E)
+space of functions f in CVk(Ω,E) that vanish with all their deriva-
+tives when weighted at infinity 51
+CVk
+0,P (∂)(Ω,E) kernel of the linear partial differential operator P(∂)E in CVk
+0(Ω,E)
+52
+CV(Ω,E)
+space of continuous functions f∶Ω → E such that fν is bounded on
+Ω for all ν ∈ V 46
+CV0(Ω,E)
+space of functions f in CV(Ω,E) such that fν vanishes at infinity
+for all ν ∈ V 46
+CVP (∂)(Ω,E)
+kernel of the linear partial differential operator P(∂)E in CV(Ω,E)
+48
+CV0,P (∂)(Ω,E) kernel of the linear partial differential operator P(∂)E in CV0(Ω,E)
+48
+D(Ω,E)
+space of càdlàg functions f∶Ω → E 44
+E0(E)
+space of functions f in C∞([0,1],E) such that (∂k)Ef(1) = 0 59
+E{Mp}(Ω,E)
+space of ultradifferentiable functions of class {Mp} of Roumieu-type
+25
+E(Mp)(Ω,E)
+space of ultradifferentiable functions of class (Mp) of Beurling-type
+25
+FG(U,E)
+space of functions f∶U → E such that for every e′ ∈ G there is
+fe′ ∈ F(Ω) with T K
+m(fe′)(x) = (e′ ○ f)(m,x) for all (m,x) ∈ U 76
+FVG(U,E)lb
+lb-restriction space 92
+FVG(U,E)sb
+sb-restriction space 89
+FV(Ω,E)σ
+space of functions f∶Ω → E such that e′ ○ f ∈ FV(Ω) for all e′ ∈ E′
+28
+FV(Ω,E)κ
+space of functions f in FV(Ω,E)σ such that Rf(B○
+α) is relatively
+compact in FV(Ω) for all α ∈ A 29
+FV(Ω,E)
+space of functions in F(Ω,E) with a weighted graph topology in-
+duced by the family of weights V 22
+AP(Ω,E)
+subspace of functions with additional properties in
+EΩ 22
+APFV(Ω,E) the space AP(Ω,E) with an emphasis on the depen-
+dence on FV(Ω) 22
+∣f∣j,m,α
+seminorms applied to f inducing the weighted graph
+topology on FV(Ω,E) 22
+∣f∣FV(Ω),j,m,αthe seminorm ∣f∣j,m,α applied to f with an emphasis
+on the dependence on FV(Ω) 22
+F(Ω,E)
+the intersection AP(Ω,E) ∩ (⋂m∈M domT E
+m) 22
+F(Ω)
+the space F(Ω,K) 22
+FV(Ω)
+the space FV(Ω,K) 22
+Nj,m(f)
+the set {T E
+m(f)(x)νj,m(x) ∣ x ∈ ωm} 22
+Fν(Ω,E)
+the space FV(Ω,E) with V = (ν) 84
+Fεν(Ω,E)
+the space of all functions S(u) s.t. u ∈ F(Ω)εE and u(B○F (Ω)′
+Fν(Ω) ) is
+bounded in E 84
+H∞(Ω,E)
+space of bounded holomorphic functions f∶Ω → E 84
+Lp(X,µ)
+space of equivalence classes of p-integrable functions f∶X → K w.r.t.
+the measure µ 142
+λ∞(A,E)
+Köthe space 42
+
+150
+List of Symbols
+ℓV(Ω,E)
+space of functions f in EΩ such that fν is bounded on Ω for all
+ν ∈ V 42
+M(Ω,E)
+space of meromorphic functions f∶Ω → E 81
+OM(Rd,E)
+multiplier space for the Schwartz space 25
+O(Ω,E)
+space of holomorphic functions f∶Ω → E 20
+s(Ω,E)
+space of sequences (xk) in E such that the sequence (xk(1+∣k∣2)j/2)
+is bounded in E for all j ∈ N 43
+Sµ(Rd,E)
+Beurling–Björck space 55
+(γ)
+property of µ 55
+S(Rd,E)
+Schwartz space 25
+XΩ
+space of maps f∶Ω → X 17
+Maps
+χK
+characteristic function of a set K ⊂ Ω 17
+∂
+Ef
+Cauchy–Riemann operator applied to an E-valued function f 20
+∆∗f
+the jump function of a càdlàg function f 139
+δx
+point evaluation functional f ↦ f(x) 21
+Hn
+n-th Hermite polynomial 128
+hn
+n-th Hermite function 128
+J
+the canonical injection E → E′⋆, x �→ [e′ ↦ e′(x)] 30
+∣µ∣
+total variation of a K-valued measure µ 142
+(∂β)Ef
+β-th partial derivative of an E-valued function f 19
+∂βf
+β-th partial derivative (∂β)Kf of a K-valued function
+f 20
+(∂n
+C)Ef
+n-th complex derivative (∂n
+C)Ef of an E-valued function f 20
+f (n)
+n-th complex derivative (∂n
+C)Cf of a C-valued func-
+tion f 20
+Rf
+the map E′ → F(Ω), e′ ↦ fe′, for given f ∈ FG(U,E) 76
+Rf
+the map E′ → FV(Ω), e′ ↦ e′ ○ f, for given f ∈ FV(Ω,E)σ 29
+Rt
+f
+the map FV(Ω)′ → E′⋆, f ′ �→ [e′ ↦ f ′(Rf(e′))], for given f ∈
+FV(Ω,E)σ 29
+S
+the map F(Ω)εE → F(Ω,E), u �→ [x ↦ u(δx)] 21
+SF(Ω)
+the map S with an emphasis on the dependence on
+F(Ω) 21
+T1εT2
+ε-product of the continuous linear operators T1 and T2 18
+T E
+m,x
+the map f ↦ T E
+m(f)(x) 23
+Θ
+the linear injection F ⊗ E → FεE 18
+V
+family of weight functions 22
+(V∞)
+vanishing at infinity condition on the family of weight
+functions 34
+Miscellaneous
+(DN)
+property of a Fréchet space 67
+(Ω)
+property of a Fréchet space 67
+(PA)
+property of a PLS-space 67
+
+Index
+almost norming 89
+B-complete 78
+BC-space 78
+Beurling–Björck space 55
+Blaschke’s convergence theorem 110
+Bloch type space 87
+Borel–Ritt theorem 73
+Br-complete 78
+càdlàg function 44
+Cauchy continuous 38
+Cauchy–Riemann operator 20
+coefficient functional 118
+completely regular 37
+consistent family 60
+continuously partially differentiable 19
+convex compactness property (ccp) 18
+metric (metric ccp) 18
+determine boundedness 78
+disk 18
+dom-space 23
+ε-compatible 21
+ε-into-compatible 21
+ε-product 17
+equicontinuous basis 118
+E-valued weak FV-function 28
+expansion operator 118
+exponential type 51
+family of weight functions 22
+C1-controlled 34
+directed 23
+locally bounded 52
+locally bounded away from zero 26, 48
+Favard space 43
+fix the topology 89, 92
+Fourier expansion 127
+Fourier transformation 55, 74
+generalised Gelfand space 109
+generalised Schwartz space 93
+generator 23
+consistent 23
+strong 23
+hat function 125
+Hermite function 128
+Hermite polynomial 127
+Hölder continuous 45
+holomorphic 20
+infra-exponential type 51
+injective tensor product 18
+jump function 139
+Köthe matrix 42
+Köthe space 42
+kR-space 37
+k-space 37
+K-valued measure 142
+lb-restriction space 92
+lcHs 17
+local closure 18
+local limit point 18
+locally closed 18
+locally complete 18
+multiplier space 25
+Nachbin-family 46
+Pettis-integrable 143
+Pettis-integral 143
+Pettis-summable 144
+PLS-space 67
+positive measure 142
+projective tensor product 70
+restriction space 76
+Riesz–Markov–Kakutani theorem 59
+sb-restriction space 89
+Schauder basis 118
+Schauder decomposition 118
+151
+
+152
+Index
+Schauder hat function 125
+Schwartz space 25
+semiflow 43
+separating subspace 17
+set of uniqueness 76
+strict topology 53, 114
+strong family 60
+topological basis 118
+total variation 142
+ultradifferentiable 25
+uniformly discrete metric space 42
+vanish at infinity in the weighted topol-
+ogy w.r.t. (π,K) 40
+weakly integrable 143
+weakly measurable 143
+weakly summable 144
+weak-strong principle 75
+Wolff’s theorem 114
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