Weighted spaces of vector-valued functions and the ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon$$\end{document}-product

We introduce a new class FV(Ω,E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {FV}(\Omega ,E)$$\end{document} of weighted spaces of functions on a set Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega$$\end{document} with values in a locally convex Hausdorff space E which covers many classical spaces of vector-valued functions like continuous, smooth, holomorphic or harmonic functions. Then we exploit the construction of FV(Ω,E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {FV}(\Omega ,E)$$\end{document} to derive sufficient conditions such that FV(Ω,E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {FV}(\Omega ,E)$$\end{document} can be linearised, i.e. that FV(Ω,E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {FV}(\Omega ,E)$$\end{document} is topologically isomorphic to the ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon$$\end{document}-product FV(Ω)εE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {FV}(\Omega )\varepsilon E$$\end{document} where FV(Ω):=FV(Ω,K)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {FV}(\Omega ):=\mathcal {FV}(\Omega ,\mathbb {K})$$\end{document} and K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {K}$$\end{document} is the scalar field of E.


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 FV(Ω) be a locally convex Hausdorff space of functions from a non-empty set Ω to a field whose topology is generated by a family V of weight functions and E be a locally convex Hausdorff space. The -product of FV(Ω) and E is defined as the space of linear continuous operators equipped with the topology of uniform convergence on equicontinuous subsets of the dual FV(Ω) � which itself is equipped with the topology of uniform convergence on absolutely convex compact subsets of FV(Ω) . Suppose that the point-evaluation functionals x , x ∈ Ω , belong to FV(Ω) � and that there is a locally convex Hausdorff space FV(Ω, E) of E-valued functions on Ω such that the map is well-defined. The main question we want to answer reads as follows. When is FV(Ω) E a linearisation of FV(Ω, E) , i.e. when is S a topological isomorphism?
In [1][2][3] 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 CV 0 (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 a topological isomorphism. Schwartz answers this question for several weighted spaces of k-times continuously partially differentiable on Ω = ℝ d like the Schwartz space in [31,32] for quasi-complete E with regard to vector-valued distributions. Grothendieck treats the question in [15], mainly for nuclear FV(Ω) and complete E. In [19][20][21] 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 [27]. For holomorphic functions on open subsets of infinite dimensional spaces a positive answer is given in [9] by Dineen. Bonet, Frerick and Jordá show in [6] that S is a topological isomorphism for certain closed subsheafs of the sheaf C ∞ (Ω, E) of smooth functions on an open subset Ω ⊂ ℝ d 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.
In [6,13,14] linearisation is used by Bonet, Frerick, Jordá and Wengenroth to derive results on extensions of vector-valued functions and weak-strong principles. Another application of linearisation is within the field of partial differential equations. Let P( ) be an elliptic linear partial differential operator with constant coefficients and C ∞ (Ω) ∶= C ∞ (Ω, ) . Then is surjective by [16,Corollary 10.6.8,p. 43] and [16,Corollary 10.8.2,p. 51]. Due to [18,Satz 10.24,p. 255], the nuclearity of C ∞ (Ω) and the topological isomorphism C ∞ (Ω, E) ≅ C ∞ (Ω) E for locally complete E, we immediately get the surjectivity of for Fréchet spaces E where P( ) E is the version of P( ) for E-valued functions. Thanks to the splitting theory of Vogt for Fréchet spaces and of Bonet and Domański for PLS-spaces we even have that P(   [10, Corollary 4.8, p. 1116] and for more details on the properties (DN), (Ω) and (PA) see [5,28].
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. This new approach is used in [24] to lift series representations from scalar-valued functions to vector-valued functions. Let us outline the content of this paper. We begin with some notation and preliminaries in Sect. 2 and introduce in Sect. 3 the spaces of functions FV(Ω, E) as subspaces of sections of domains of linear operators T E on E Ω having a certain growth given by a family of weight functions V . These spaces cover many examples of classical spaces of functions appearing in analysis like the mentioned ones. Then we exploit the structure of our spaces to describe sufficient conditions, which we call consistency and strength, on the interplay of the pairs of operators (T E , T ) and the map S as well as the spaces FV(Ω) and E such that S ∶ FV(Ω) E ≅ FV(Ω, E) becomes a topological isomorphism in our main Theorem 14. Looking at the pair of partial differential operators (P( ) E , P( )) considered above, these conditions allow us to express P( ) E as P( Hence it becomes obvious that the surjectivity of P( ) E is equivalent to the surjectivity of P( ) id E . This is used in [23,26] in the case of the Cauchy-Riemann operator P( ) = on spaces of smooth functions with exponential growth.

Notation and preliminaries
We equip the spaces ℝ d , d ∈ ℕ , and ℂ with the usual Euclidean norm | ⋅ | . Furthermore, for a subset M of a topological space X we denote the closure of M by M and the boundary of M by M . 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.
By E we always denote a non-trivial locally convex Hausdorff space over the field = ℝ or ℂ equipped with a directed fundamental system of seminorms (p ) ∈ and, in short, we write that E is an lcHs. If E = , then we set (p ) ∈ ∶= {| ⋅ |}. For details on the theory of locally convex spaces see [12,17] or [28].
By X Ω we denote the set of maps from a non-empty set Ω to a non-empty set X, by K 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 L(F, E) the space of continuous linear operators from F to E where F and E are locally convex Hausdorff spaces. If E = , we just write F � ∶= L(F, ) for the dual space and G • for the polar set of G ⊂ F . If F and E are (linearly) topologically isomorphic, we write F ≅ E . We denote by L t (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 = , on the absolutely convex, compact subsets of F if t = and on the precompact (totally bounded) subsets of F if t = . We use the symbols t(F � , F) for the corresponding topology on F ′ and t(F) for the corresponding bornology on F. The so-called -product of Schwartz is defined by where L(F � , E) is equipped with the topology of uniform convergence on equicontinuous subsets of F ′ . This definition of the -product coincides with the original one by Schwartz [32, Chap. I, Sect. 1, Définition, p. 18]. It is symmetric which means that F E ≅ E F . In the literature the definition of the -product is sometimes done the other way around, i.e. E F is defined by the right-hand side of (1) but due to the symmetry these definitions are equivalent and for our purpose the given definition is more suitable. If we replace F ′ by F ′ , we obtain Grothendieck's definition of the -product and we remark that the two -products coincide if F is quasi-complete because then F � = F � holds. However, we stick to Schwartz' definition. For more information on the theory of -products see [17,18].
The sufficient conditions for the surjectivity of the map S ∶ FV(Ω) E → FV(Ω, 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 vector space E D ∶= ⋃ n∈ℕ nD becomes a normed space if it is equipped with the gauge functional of D as a norm (see [17, p. 151]). The space E is called locally complete if E D is a Banach space for every closed disk D ⊂ E (see [17,10.2.1 Proposition,p. 197]). Moreover, a locally convex Hausdorff space is locally complete if and only if it is convenient by [22, 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 [17, p. 58 [7,35]. In addition, we remark that every semi-Montel space is semireflexive by [17, 11.5.1 Proposition, p. 230] and every semi-reflexive locally convex Hausdorff space is quasi-complete by [30, Chap. IV, 5.5, Corollary 1, p. 144] and these implications are strict as well. Summarizing, we have the following diagram of strict implications: Since weighted spaces of continuously partially differentiable vector-valued functions will serve as our standard examples, we recall the definition of the spaces if n ≠ 0 and the right-hand side exists in E for every x ∈ Ω . Further, we define if the right-hand side exists in E for every x ∈ Ω.

The "-product for weighted function spaces
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 E Ω 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 we can identify FV(Ω, E) with FV(Ω) E topologically. This is deeply connected with the interplay of the pair of operators (T E m , T m ) with the map S from the introduction (see Definition 6). In our main theorem we give sufficient conditions such that FV(Ω, E) ≅ FV(Ω) E holds (see Theorem 14). We start with the well-known example C k (Ω, E) of k-times continuously partially differentiable E-valued functions to motivate our definition of FV(Ω, E).
compact-open topology, i.e. the topology given by the seminorms for compact K ⊂ Ω and ∈ . The usual topology on the space C k (Ω, E) of k-times continuously partially differentiable functions is the graph topology generated by the partial derivative operators e. the topology given by the seminorms The same topology is induced by the directed systems of seminorms given by for compact K ⊂ Ω , m ∈ ℕ 0 , m ≤ k , and ∈ and may also be seen as a weighted topology induced by the family ( K ) of characteristic functions of the compact sets K ⊂ Ω by writing This topology is inherited by linear subspaces of functions having additional properties like being holomorphic or harmonic.
We turn to the weight functions which we use to define a kind of weighted graph topology.
Definition 2 (Weight function) Let J be a non-empty set and ( m ) m∈M a family of non-empty sets. We call V ∶= ( j,m ) j∈J,m∈M a family of weight functions From the structure of Example 1 we arrive at the following definition of the weighted spaces of vector-valued functions we want to consider. 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 E Ω or C 0 (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 , for x ∈ Ω and their continuity play a big role, we introduce the following definition.

Remark 5
a) It is easy to see that then T m,x ∈ FV(Ω) � for every m ∈ M and x ∈ m . Indeed, for m ∈ M and x ∈ m there exists j ∈ J such that j,m (x) > 0 by (2), implying for every f ∈ FV(Ω) that In particular, this implies x ∈ FV(Ω) � for all x ∈ Ω if there is m ∈ M such that m = Ω and T m = id Ω. c) The system of seminorms (|f | j,m, ) j∈J,m∈M, ∈ is directed if the family of weight functions V is directed, i.e.
since the system (p ) ∈ of E is already directed.
More precisely, T m,x in b) means the restriction of T m,x to FV(Ω) and the term u(T m,x ) is well-defined by Remark 5 b). Consistency will guarantee that the map S ∶ FV(Ω) E → FV(Ω, E) is a well-defined topological isomorphism into, i.e. to its range, and strength will help us to prove its surjectivity under some additional assumptions on FV(Ω) and E. Let us come to a lemma which describes the topology of FV(Ω) E in terms of the operators T m with m ∈ M . It was the motivation for the definition of consistency and allows us to consider FV(Ω) E as a topological subspace of FV(Ω, E) via S, assuming consistency. for j ∈ J , m ∈ M and ∈ gives the topology on FV(Ω) E (here it is used that the system of seminorms (| ⋅ | j,m ) of FV(Ω) is directed). As every u ∈ FV(Ω) E is continuous on B • j,m , we may replace B • j,m by a (FV(Ω) � , FV(Ω))-dense subset. Therefore we obtain thus q j,m, (u) ≤ ‖u‖ j,m, . On the other hand, we derive ◻ Let us turn to a more general version of Example 1, namely, to weighted spaces of k-times continuously partially differentiable functions and kernels of partial differential operators in these spaces.
This is a special case of a)(i) with k = ∞ , Ω = ℝ d , J = {1} and We set and obtain the (topological) subspace of CV k (Ω, E) given by Choosing AP(Ω, E) ∶= ker P( ) E , we see that this is also a dom-space by a). If P( ) E is the Cauchy-Riemann operator or the Laplacian, we obtain the weighted space of holomorphic resp. harmonic functions.
We note that Example 8 a)(ii) covers spaces of ultradifferentiable functions. Let us show that the generator of these spaces is strong and consistent. In order to obtain consistency for their generator we have to restrict to directed families of weights which are locally bounded away from zero on Ω , i.e. This condition on V k guarantees that the map I ∶ CV k (Ω) → CW k (Ω) , f ↦ f , is continuous which is needed for consistency.

Proposition 9
Let E be an lcHs, k ∈ ℕ ∞ , V k be a directed family of weights which is locally bounded away from zero on an open set Ω ⊂ ℝ d . The generator of (CV k , E) resp. (CV k P( ) , E) from Example 8 is strong and consistent if CV k (Ω) resp. CV k P( ) (Ω ) is barrelled.

Proof We recall the definitions from Example 8. We have
for all m ∈ ℕ 0 and the same with instead of E. The family (T E m , T m ) m∈ℕ 0 is a strong generator for (CV k , E) because for all e � ∈ E � , f ∈ CV k (Ω, E) and m ∈ ℕ 0 due to the linearity and continuity of e � ∈ E � . In addition, e � •f ∈ ker P( ) for all e � ∈ E � and f ∈ CV k P( ) (Ω, E) which implies that (T E m , T m ) m∈ℕ 0 is also a strong generator for (CV k P( ) , E). For consistency we need to prove that for all u ∈ CV k (Ω) E resp. u ∈ CV k P( ) (Ω) E . This follows from the subsequent Proposition 10 b) since FV(Ω) = CV k (Ω) resp. FV(Ω) = CV k P( ) (Ω) is barrelled and V k locally bounded away from zero on Ω . Thus (T E m , T m ) m∈ℕ 0 is a consistent generator for (CV k , E) . In addition, we have with P( ) E from Example 8 d) that  The Schwartz space from Example 8 c) can also be topologized by integral operators instead of partial derivative operators. Let f ∈ S(ℝ d , E) . If E is sequentially complete, then f •h n is Pettis-integrable on ℝ d for every n = (n k ) ∈ ℕ d 0 by [24,4.8 Proposition,p. 15] , is the nth Hermite function with Thus the nth Fourier coefficient of f given by the Pettis-integral is defined if E is sequentially complete.

Example 11
Let E be a sequentially complete lcHs and equip S(ℝ d , E) with the topology generated by the seminorms for j ∈ ℕ and ∈ . S(ℝ d , E) equipped with this topology is a dom-space with for j ∈ ℕ whose topology coincides with the one from Example 8 c) by [24,4.9 Theorem,p. 16]. Further, by the same theorem the generator (F E , F ) of (S, E) is strong and consistent.
Among others, the techniques of the present paper are used in [24] to deduce the series expansion and to show that for every f ∈ FV(Ω, E) , j ∈ J , m ∈ M and ∈ with the set N j,m (f ) from Definition 3.
Proof Let f ∈ FV(Ω, E) . We have e � •f ∈ F(Ω) for every e � ∈ E � since (T E m , T m ) m∈M is a strong generator. Moreover, we have for every j ∈ J and m ∈ M . We note that N j,m (f ) is bounded in E by Definition 3 and thus weakly bounded, implying that the right-hand side of (4) is finite. Hence we conclude f ∈ FV(Ω, E) . Further, we observe that for every j ∈ J , m ∈ M and ∈ due to [28,Proposition 22.14,p. 256].

a) FV(Ω) is a semi-Montel space. b) E is a semi-Montel space. c)
Let us apply our preceding results to our weighted spaces of k-times continuously partially differentiable functions on an open set Ω ⊂ ℝ d with k ∈ ℕ ∞ .

Example 16
Let E be an lcHs, k ∈ ℕ ∞ , V k be a directed family of weights which is locally bounded away from zero on an open set Ω ⊂ ℝ d .
Proof The generator of (CV k , E) and (CV k P( ) , E) is strong and consistent by Proposition 9. From Theorem 14 (ii) we deduce part a) and b) and from (i) part c) and d). The precompactness of N K,j,m (f ) implies that there exists a finite set P ⊂ E such that N K,j,m (f ) ⊂ P + U . Hence we conclude which means that N j,m (f ) is precompact. Since E is quasi-complete, N j,m (f ) is relatively compact as well by [17, 3.5 [18, (9), p. 236] (without a proof) that CW k (Ω, E) ≅ CW k (Ω) E for k ∈ ℕ ∞ and quasi-complete E. For k = ∞ we even have CW ∞ (Ω, E) ≅ CW ∞ (Ω) E for locally complete E by [6, p. 228]. Our technique allows us to generalise the first result and to get back the second result.
Proof We recall from Example 8 b) that W k is the family of weights given by and K is the characteristic function of K. We already know that the generator for (CW k , E) and (CW k P( ) , E) is strong and consistent by Proposition 9 because W k is locally bounded away from zero on Ω , CW k (Ω) and its closed subspace CW k P( ) (Ω) are Fréchet spaces. Let f ∈ CW k (Ω, E) , K ⊂ Ω be compact, m ∈ ℕ 0 and consider  for m ∈ ℕ 0 and ∈ if k = ∞ . We prepare the proof of consistency of its generator. We write C ext (Ω, E) for the space of functions f ∈ C(Ω, E) which have a continuous extension to Ω and set C ext (Ω) ∶= C ext (Ω, ) . We equip C ext (Ω) with the topology of uniform convergence on compact subsets of Ω.
Proof From Proposition 10 a) we derive that •T ∈ C(Ω, FV(Ω) � ) . Let x ∈ Ω and (x n ) a sequence in Ω with x n → x . Then ( x n •T) is a sequence in FV(Ω) � and in for every f ∈ FV(Ω) , which implies that ( x n •T) converges to ext x •T pointwise in f because T(f ) ∈ C ext (Ω) . As a consequence of the Banach-Steinhaus theorem we get ( ext x •T) ∈ FV(Ω) � and the convergence in FV(Ω) � . ◻

Example 20
Let E be an lcHs, k ∈ ℕ ∞ and Ω ⊂ ℝ d open and bounded. Then Proof The generator coincides with the one of Example 8 a)(i). Due to Proposition 10 we have S(u) ∈ C k (Ω, E) and for all u ∈ C k (Ω) E since C k (Ω) is a Banach space if k < ∞ and a Fréchet space if k = ∞ , in particular, both are barrelled. As a consequence of Proposition 19 with T = ( ) for ∈ ℕ d 0 , | | ≤ k , we obtain that ( ) E S(u) ∈ C ext (Ω, E) for all u ∈ C k (Ω) E . Hence the generator is consistent. It is easy to check that it is strong too.
We denote by f the continuous extension of ( ) E f on the compact metrisable set Ω . The set is relatively compact and metrisable since it is a subset of a finite union of the compact metrisable sets f (Ω) like in Example 18. Due to Theorem 14 (iii) we obtain our statement as E has metric ccp. ◻