Weighted vector-valued functions and the $\varepsilon$-product

We introduce a new class $\mathcal{FV}(\Omega,E)$ of spaces of weighted functions on a set $\Omega$ 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 $\mathcal{FV}(\Omega,E)$ to derive sufficient conditions such that $\mathcal{FV}(\Omega,E)$ can be linearised, i.e. that $\mathcal{FV}(\Omega,E)$ is topologically isomorphic to the $\varepsilon$-product $\mathcal{FV}(\Omega)\varepsilon E$ where $\mathcal{FV}(\Omega):=\mathcal{FV}(\Omega,\mathbb{K})$ and $\mathbb{K}$ is the scalar field of $E$.


Introduction
This work is dedicated to a classical topic, namely, the linearisation of spaces of weighted vector-valued functions. The setting we are interested in is the following. Let F V(Ω) be a locally convex Hausdorff space of functions from a non-empty set Ω to a field K whose topology is generated by a family V of weight functions on Ω and E be a locally convex Hausdorff space. The ε-product of F V(Ω) and E is defined as the space of linear continuous operators F V(Ω)εE ∶= L e (F V(Ω) ′ κ , E) equipped with the topology of uniform convergence on equicontinuous subsets of F V(Ω) ′ which itself is equipped with the topology of uniform convergence on absolutely convex compact subsets of F V(Ω). Suppose that there is a locally convex Hausdorff space F V(Ω, E) of E-valued functions on Ω such that the map is well-defined where δ x , x ∈ Ω, is the point-evaluation functional. The main question we want to answer reads as follows. When is F V(Ω)εE a linearisation of F V(Ω, E), i.e. when is S a topological isomorphism?
In [2], [3] and [4] 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 spaces of weighted k-times continuously partially differentiable on R d like the Schwartz space in [36] and [37] for quasi-complete E with regard to vector-valued distributions. Grothendieck treats the question in [18], mainly for nuclear F V(Ω) and complete E. In [24], [25] and [26] Komatsu gives a positive answer for ultradifferentiable functions of Beurling or Roumieux 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 [32]. For holomorphic functions on open subsets of infinite dimensional spaces a positive answer is given in [11] by Dineen. Bonet, Frerick and Jordá show in [8] 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.
Our goal is to give a unified approach to linearisation which is able to handle new examples and covers the already known examples. This new approach is used in [30] to generalise the extension results of [8], [16] and [17] and to lift series representations from scalar-valued functions to vector-valued functions in [31]. Let us outline the content of this paper. We begin with some notation and preliminaries and introduce in the third section the spaces of functions F V(Ω, E) as sections of domains and kernels of linear operators T E on E Ω having certain growth conditions 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 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 a topological isomorphism into (see Theorem 3.7). In our main Theorem 3.14 and its Corollaries 3.15, 3.16 and 3.17 we give several sufficient conditions on the pairs of operators (T E , T K ) and spaces involved such that F V(Ω)εE ≅ F V(Ω, E) holds via S. In the fourth section we treat the question which properties of functions can be described by our pairs of operators (T E , T K ) and when do they fulfil our sufficient conditions. We close this work with many examples in the fifth section.

Notation and Preliminaries
We equip the spaces R d , d ∈ N, and C with the usual Euclidean norm ⋅ , denote by B r (x) ∶= {w ∈ R d w − x < r} the ball around x ∈ R d with radius r > 0. 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 K = R or C equipped with a directed fundamental system of seminorms (p α ) α∈A and, in short, we write E is an lcHs. If E = K, then we set (p α ) α∈A ∶= { ⋅ }. For details on the theory of locally convex spaces see [15], [22] or [33]. By X Ω we denote the set of maps from a non-empty set Ω to a non-empty set X, by χ 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 = 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 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 = 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 corresponding bornology on F . The so-called ε-product of Schwartz is defined by where L(F ′ κ , E) is equipped with the topology of uniform convergence on equicontinuous subsets of F ′ . This definition of the ε-product coincides with the original one by Schwartz [37, Chap. I, §1, Définition, p. 18]. It is symmetric which means that F εE ≅ 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 (1) 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 more information on the theory of ε-products see [22] and [23]. Further, for a disk D ⊂ F , i.e. a bounded, absolutely convex set, the vector space F D ∶= ⋃ n∈N nD becomes a normed space if it is equipped with gauge functional of D as a norm (see [22, p. 151]). The space F is called locally complete if F D is a Banach space for every closed disk D ⊂ F (see [22,10.2.1 Proposition,p. 197]).

The ε-product for weighted function spaces
In this section we introduce the space F V(Ω, E) of weighted E-valued functions on Ω as the section of domains and kernels 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 as an example for these operators we can think of the partial derivative operators. Then we treat the question whether we can identify F V(Ω, E) with F V(Ω)εE topologically. This is deeply connected with the interplay of the pair of operators (T E m , T K m ) with the map S from the introduction (see Definition 3.6). In our main theorem we give sufficient conditions such that F V(Ω, E) ≅ F V(Ω)εE holds (see Theorem 3.14).
We begin with the definition of a family of weight functions which we want to use to define a kind of weighted graph topology.
3.1. Definition (weight function). Let Ω, J, L be non-empty sets and (M l ) l∈L a family of non-empty sets. We call V ∶= ((ν j,l,m ) m∈M l ) j∈J,l∈L a family of weight functions on Ω if ν j,l,m ∶ Ω → [0, ∞) for every m ∈ M l , j ∈ J and l ∈ L and Now, the spaces we want to consider are built up in the following way.

3.2.
Definition. Let V ∶= ((ν j,l,m ) m∈M l ) j∈J,l∈L be a family of weight functions on Ω and M top ∶= ⋃ l∈L M l . Let M 0 and M r be sets, M top , M 0 and M r be pairwise disjoint and M ∶= M top ∪ M 0 ∪ M r . Let (ω m ) m∈M be a family of non-empty sets such that Ω ⊂ ω m for every m ∈ M top and T E m ∶ E Ω ⊃ dom T E m → E ωm is a linear map for every m ∈ M. We define the space of intersections ker T E m as well as The space F V(Ω, E) is a locally convex Hausdorff space due to condition (2). Since it is easier to work with a directed family of seminorms and the (continuity of the) point evaluation functionals δ , for x ∈ Ω play a big role, we make the following definition.
, for m ∈ M and x ∈ ω m . 3.4. Remark. It is easy to see that the system of seminorms ( f j,l,α ) j∈J,l∈L,α∈A is directed if (2). The next lemma describes the topology of F V(Ω)εE in terms of the operators T K m with m ∈ M top and is a preparation to consider F V(Ω)εE as a topological subspace of F V(Ω, E) under certain conditions. 3.5. Lemma. Let F V(Ω) be a dom-space. Then the following holds.
The topology of F V(Ω)εE is given by the system of seminorms defined by for j ∈ J, l ∈ L and α ∈ A.
Proof. a) For x ∈ Ω and m ∈ M top there exist l ∈ L and j ∈ J such that m ∈ M l and ν j,l,m (x) > 0 by (2) implying for every f ∈ F V(Ω) that x ∈ Ω, m ∈ M l } and B j,l ∶= {f ∈ F V(Ω) f j,l ≤ 1} for every j ∈ J and l ∈ L. We claim that acx(D j,l ) is dense in the polar B ○ j,l with respect to κ(F V(Ω) ′ , F V(Ω)). The observation by the bipolar theorem. By [22, 8.4, p. 152, 8.5, p. 156-157] the system of seminorms defined by for j ∈ J, l ∈ L and α ∈ A gives the topology on F V(Ω)εE (here it is used that the system of seminorms ( ⋅ j,l ) of F V(Ω) is directed). We may replace B ○ j,l by a κ(F V(Ω) ′ , F V(Ω))-dense subset as every u ∈ F V(Ω)εE is continuous on B ○ j,l . Therefore we obtain q j,l,α (u) = sup p α u(y) y ∈ acx(D j,l ) .
thus q j,l,α (u) ≤ u j,n,α . On the other hand, we derive For the lcHs E over K we want to define a natural E-valued version of a domspace F V(Ω) = F V(Ω, K). Let M ∶= M top ∪M 0 ∪M r be the index set associated to F V(Ω). The natural E-valued version of F V(Ω) should be a dom-space F V(Ω, E) such that the three parts of its index set coincide with the corresponding parts of M and there is a canonical relation between the families (T K m ) and (T E m ). This canoncial relation will be explained in terms of their interplay with the map  More precisely, T K m,x in (i) and (ii) means the restriction of T K m,x to F V(Ω). Consistency is our measure whether we consider a space F V(Ω, E) as a natural E-valued version of a space F V(Ω) of scalar-valued functions.
For m ∈ M 0 we get T K m,x = 0 on F V(Ω) for all x ∈ ω m and thus Hence S(u) ∈ ker T E m for every m ∈ M 0 . Furthermore, we get by Lemma 3.5 b) for every j ∈ J, l ∈ L and α ∈ A implying S(u) ∈ F V(Ω, E) and the continuity of S. Moreover, we deduce from (3) that S is injective and that the inverse of S on the range of S is also continuous.

3.8.
Remark. If J, L and A are countable, then S is an isometry with respect to the induced metrics on F V(Ω, E) and F V(Ω)εE by (3).
The basic idea for Theorem 3.7 was derived from analysing the proof of an analogous statement for weighted continuous functions by Bierstedt [3,4.2 Lemma,4.3 Folgerung, and [4, 2.1 Satz, p. 137]. Now, we try to answer the natural question. When is S surjective? A weaker concept to define a natural E-valued version of F V(Ω) will help us to answer the question. Let F V(Ω) be a dom-space. We define the vector space of E-valued weak F V-functions by Moreover, for f ∈ F V(Ω, E) σ we define the linear map Next, we give a sufficient condition for the inclusion F V(Ω, E) ⊂ F V(Ω, E) σ by means of the family (T E m , T K m ) m∈M . 3.9. Definition (strong). Let (T E m , T K m ) m∈M be a defining family for (F V, E). We call (T E m , T K m ) m∈M strong if the following is valid for every e ′ ∈ E ′ , f ∈ F V(Ω, E) and m ∈ M: We call (T E m , T K m ) m∈N a strong subfamiliy if (i) and (ii) are fulfilled for every m ∈ N .
for every f ∈ F V(Ω, E), j ∈ J, l ∈ L and α ∈ A where for every j ∈ J and l ∈ L. Further, we observe that for every j ∈ J, l ∈ L and α ∈ A where the first equality holds due to [33,Proposition 22.14,p. 256]. In particular, we obtain that N j,l (f ) is bounded in E and thus weakly bounded implying that the right-hand side of (5) is finite. Hence we conclude f ∈ F V(Ω, E) σ . Now, we phrase some sufficient conditions for F V(Ω, E) ⊂ F V(Ω, E) κ to hold which is one of the key points regarding the surjectivity of S.
3.11. Lemma. If (T E m , T K m ) m∈M is a strong family for (F V, E) and one of the following conditions is fulfilled, There are a set X, a family K of sets and a map π∶ Ω × M top → X such that ⋃ K∈K K ⊂ X and the functions of F V(Ω, E) vanish at infinity in the weighted topology with respect to (π, K), i.e. every f ∈ F V(Ω, E) fulfils: . By virtue of Lemma 3.10 we already have f ∈ F V(Ω, E) σ . a) For every j ∈ J, l ∈ L and α ∈ A we derive from is bounded and thus relatively compact in the semi-Montel space F V(Ω). b) It follows from (5) that R f ∈ L(E ′ γ , F V(Ω)). Further, the polar B ○ α is relatively compact in E ′ γ for every α ∈ A by the Alaoğlu-Bourbaki theorem. The continuity of R f implies that R f (B ○ α ) is relatively compact as well.
c) Let j ∈ J and l ∈ L. The set K ∶= N j,l (f ) is bounded in E by (4). If E is semi-Montel or Schwartz, we deduce that K is already precompact in E since it is relatively compact if E is semi-Montel resp. by [22, 10.4 Concerning d), in all examples we consider later on we have to assume that K is closed under taking finite unions (see Proposition 4.2). The most common case is that K consists of the compact subsets of Ω and π is the projection on X ∶= Ω. But we consider other examples in Example 5.9 as well.
3.12. Remark. Let F V(Ω, E) be a dom-space, Ω be a topological Hausdorff space, M l be finite for every l ∈ L, every ν ∈ V be bounded on the compact subsets of Ω, every f ∈ F V(Ω, E) fulfil (6) with K ∶= {K ⊂ Ω K compact} and π be the projection on X ∶= Ω. If T E m (f ) ∈ C(Ω, E) for every f ∈ F V(Ω, E) and m ∈ M top , then N π⊂K,j,l (f ) is precompact in E for every f ∈ F V(Ω, E), K ∈ K, j ∈ J and l ∈ L.
Proof. Let f ∈ F V(Ω, E), K ∈ K, j ∈ J and l ∈ L. Writing we see that we only have to prove that the sets T E m (f )(K)ν j,l,m (K) are precompact since N π⊂K,j,l (f ) is a finite union of these sets. But this is a consequence of the proof of [2, §1, 16. Lemma, p. 15].

Let us turn to sufficient conditions for
Proof. Due to Theorem 3.7 we only have to show that S is surjective. We equip J (E) with the system of seminorms given by for every α ∈ A. Let f ∈ F V(Ω, E). We consider the dual map R t f and claim that is absolutely convex and relatively compact implying that K α is absolutely convex and compact in F V(Ω) by [22, 6.2 As for all α ∈ A, we gain that (R t f (f ′ τ )) τ is a Cauchy net in the complete space J (E). Hence it has a limit g ∈ J (E) which coincides with R t f (f ′ ) since .

as vector spaces) and we gain
In particular, we get the following corollaries as special cases of Theorem 3.14.
Proof. Follows from Lemma 3.11 a) and Theorem 3.14 with Property 3.13 a).
Proof. We observe that acx(N j,l (f )) is absolutely convex and compact in the semi-Montel space E by [22, 6.2.1 Proposition, p. 103] and [22, 6.7.1 Proposition, p. 112] for every f ∈ F V(Ω, E), j ∈ J and l ∈ L. Our statement follows from Lemma 3.11 c) and Theorem 3.14 with Property 3.13 d).
3.17. Corollary. Let E be quasi-complete, (T E m , T K m ) m∈M a strong, consistent family for (F V, E) and the conditions of Remark 3.12 be fulfilled. Then F V(Ω, E) ≅ F V(Ω)εE.
Proof. Let f ∈ F V(Ω, E). The set N π⊂K,j,l (f ) is precompact in E due to Remark 3.12 for every K ∈ K, j ∈ J and l ∈ L. It follows from the proof of Lemma 3.11 d) that N j,l (f ) is precompact in E. Since E is quasi-complete, N j,l (f ) is relatively compact as well by [22,3.5.3 Proposition,p. 65]. This implies that K ∶= acx(N j,l (f )) is absolutely convex and compact by [41, 9-2-10 Example, p. 134] because E is quasicomplete. Our statement follows from Lemma 3.11 d) and Theorem 3.14 with Property 3.13 d).
We close this section by phrasing some sufficient conditions in Proposition 3.19 such that F V(Ω, E) ≅ F V(Ω)εE passes on to topological subspaces which will simplify our proofs when considering subspaces.
and G(Ω) a locally convex Hausdorff space of functions from Ω to K such that the inclusion G(Ω) ⊂ F V(Ω) holds topologically, then the conditions (i) and (ii) of the consistency-Definition 3.6 are satisfied for every u ∈ G(Ω)εE.
is a vector space of functions from Ω to E such that G(Ω, E) ⊂ F V(Ω, E) as vector spaces, then the conditions (i) and (ii) of the strength-Definition 3.9 are satisfied for every f ∈ G(Ω, E).
Proof. We start with a). Since F V(Ω) is a dom-space and G(Ω) ⊂ F V(Ω) holds topologically, we obtain that δ x ∈ G(Ω) ′ for every x ∈ Ω. Furthermore, every com- In addition, the restriction of every equicontinuous subset of F V(Ω) ′ to GV(Ω) is an equicontinuous subset of GV(Ω) ′ implying the continuity of the embedding GV(Ω)εE ↪ F V(Ω)εE. Hence we observe that the restriction u F V(Ω) ′ ∈ F V(Ω)εE for every u ∈ GV(Ω)εE and for every x ∈ Ω. Thus we have S(u) = S(u F V(Ω) ′ ) and u F V(Ω) ′ ∈ F V(Ω)εE for every u ∈ GV(Ω)εE. Therefore the conditions (i) and (ii) of the consistency-Definition are satisfied for every , then the conditions (i) and (ii) of the strength-Definition are satisfied for every and one of the following conditions is satisfied: and the subfamily (T E m , T K m ) m∈Mg is strong and consistent for (GV, E).
Proof. By Remark 3.18 (T E m , T K m ) m∈M∪Mg is a strong, consistent family for (GV, E). Further, we get GV(Ω, E) ⊂ GV(Ω, E) κ from Lemma 3.11 b), c) resp. d) in case (i) because GV(Ω, E) ⊂ F V(Ω, E) and from Lemma 3.11 a) in case (ii) because GV(Ω) ⊂ F V(Ω) and closed subspaces of semi-Montel spaces are semi-Montel again. If one of the Properties 3.13 a) or d) is fulfilled for (F V, E), then it is also valid for (GV, E) due to the inclusion GV(Ω, E) ⊂ F V(Ω, E). Hence Theorem 3.14 yields the statement.

Strong and consistent families
This section is dedicated to the properties of functions which can be described by defining families and answering the question when these defining families are strong and consistent. This is done in a quite general way so that we are not tied to certain spaces and have to redo our argumentation if we consider the same subfamily (T E m , T K m ) for two different spaces of functions. Among the properties of functions that can be described by strong, consistent families are vanishing at infinity by Proposition 4.2, continuity by Proposition 4.3, Cauchy continuity by Proposition 4.6, uniform continuity by 4.8, continuous extendability by Proposition 4.10, differentiability by Proposition 4.12 and purely algebraic properties of a function like linearity by Proposition 4.14. 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. We start with the properties we want to describe.
. c) Cauchy continuity: Let cc ∈ M r , Ω be a metric space and f) differentiability on a subset: Let X be a vector space over the field K 1 = R or C and ω ⊂ Ω ⊂ X. Let v ∈ X be such that for every . h) additivity: Let a ∈ M 0 , Ω be a vector space and set , Ω be a vector space over K and set with m ∈ {c, cc, ext, ∂ v , ω 0 , a, h} and the subfamily (T E m , T K m ) m∈Muc are strong subfamilies due to simple calculations and the linearity and (uniform) continuity of every e ′ ∈ E ′ . Therefore we turn our attention to the question of consistency and in a) to strength as well.

4.2.
Proposition (vanishing at infinity w.r.t. to (π, K)). If (T E m , T K m ) m∈Mtop is a strong resp. consistent subfamily for (F V, E) and K is closed under taking finite unions, then (T E ∞ , T K ∞ ) is a strong resp. consistent subfamily for (F V, E).
Proof. First, we consider consistency. We set B j,l ∶= {f ∈ F V(Ω) f j,l ≤ 1} for j ∈ J and l ∈ L. Let u ∈ F V(Ω)εE. The topologies σ(F V(Ω) ′ , F V(Ω)) and κ(F V(Ω) ′ , F V(Ω)) coincide on the equicontinuous set B ○ j,l and we deduce that the restriction of u to B ○ j,l is σ(F V(Ω) ′ , F V(Ω))-continuous. Let ε > 0, j ∈ J, l ∈ L, α ∈ A and set B α,ε ∶= {x ∈ E p α (x) < ε}. Then there are a finite set N ⊂ F V(Ω) and η > 0 such that for every f ∈ N . It follows from (10) and (the proof of) Lemma 3.5 b) that ∞ and the other conditions for the consistency of (T E ∞ , T K ∞ ) are obviously fulfilled. Let us consider strength. Let (T E m , T K m ) m∈Mtop be a strong subfamily for Using that (T E m , T K m ) m∈Mtop is a strong subfamily for (F V, E), it follows that sup x∈Ω, m∈M l π(x,m)∉K . The 'consistency'-part of the proof above adapts an idea in the proof of [3,4.4 Theorem, which is a special case of our proposition.
is a strong and consistent subfamily for c,x ) for x ∈ Ω. Now, we tackle the problem of the continuity of δ∶ Ω → F V(Ω) ′ κ in the proposition above and phrase our solution in a way such that it can be applied to show the consistency of the subfamily describing the continuity of partial derivatives as well. We recall that a topological space Ω is called completely regular (Tychonoff or T 3 1 2space) if for any non-empty closed subset F ⊂ Ω and x ∈ Ω∖F there is f ∈ C(Ω, [0, 1]) such that f (x) = 0 and f (z) = 1 for all z ∈ F (see [ σ is well-defined by Lemma 3.5 a) and we claim that it is continuous. If x ∈ Ω and (x τ ) τ ∈T is a net in Ω converging to x, then is continuous on Ω which proves our claim. (i) Let K ⊂ Ω be compact. Then there are j ∈ J, l ∈ L and C > 0 such that for every f ∈ F V(Ω). This means that {T K m,x x ∈ K} is equicontinuous in F V(Ω) ′ . The topologies σ(F V(Ω) ′ , F V(Ω)) and γ(F V(Ω) ′ , F V(Ω)) coincide on equicontinuous subsets of F V(Ω) ′ implying that the restriction (δ ○ T K m ) K ∶ K → F V(Ω) ′ γ is continuous by our first claim. As δ ○ T K m is continuous on every compact subset of the k R -space Ω, it follows that δ ○ T K m ∶ Ω → F V(Ω) ′ γ is continuous. (ii) There are j ∈ J, l ∈ L and C > 0 such that for every f ∈ F V(Ω). This means that {T K m,x x ∈ Ω} is equicontinuous in F V(Ω) ′ yielding to the statement like before.
The preceding lemma is just a modification of [3,4.1 Lemma,p. 198] where F V(Ω) = CV(Ω), the space of Nachbin-weighted continuous functions, and T K m = id Ω K . Next, we consider the special case of continuous, linear operators. Let (F, t) be a locally convex Hausdorff space with topology t and F ′ the dual with respect to t. Due to the Mackey-Arens theorem F = (F ′ κ ′ holds algebraically and thus δ∶ F → F ′ κ ′ κ induces a locally convex topology ς on F . This topology fulfils t ≤ ς ≤ τ (F, F ′ ). In particular, if F is a Mackey space, i.e. t = τ (F, F ′ ), then t = ς (see [37, Chap. I, §1, p. 17] where the topology ς is called γ).

Remark.
Let Ω be a locally convex Hausdorff space.
Proof. Part (i) follows directly from the definition of ς. Further, if Ω is quasibarrelled, then it has the Mackey-topology by [34, Observation 4.1.5 (a), p. 96], and, if Ω is bornological, then it is quasi-barrelled by [34, Observation 6.1.2 (c), p. 167]. Let us turn to part (ii). Let (x n ) be a sequence in Ω converging to x ∈ Ω. We observe that (δ xn ) converges to δ If Ω is normed or a semi-reflexive, metrisable space, then Ω ′ b is barrelled since it is a Banach space resp. by [22, 11.4.1 Proposition, p. 227]. The Banach-Steinhaus theorem yields our result.
Next, we turn to the problem of describing Cauchy continuity by strong and consistent families.

Proposition (Cauchy continuity). (T E cc , T K cc ) is a strong and consistent subfamily for
Proof. Let u ∈ F V(Ω)εE and (δ xn ) be a Cauchy sequence in F V(Ω) ′ κ . Then (S(u)(x n )) is a Cauchy sequence in E since u is uniformly continuous and u(δ xn ) = S(u)(x n ). Hence we conclude S(u) ∈ dom T E cc . The remaining part is obvious. We write CC(Ω) γ resp. CC b (Ω) for the space of scalar-valued Cauchy continuous functions equipped with the topology of uniform convergence on precompact sets resp. the space of scalar-valued bounded Cauchy continuous functions equipped with the topology of uniform convergence on Ω.

4.7.
Lemma. Let F V(Ω) be a dom-space, Ω a metric space and m ∈ M top with T K m (F V(Ω)) ⊂ CC(Ω). Then the sequence (T K m,xn ) is Cauchy in F V(Ω) ′ γ for every Cauchy sequence (x n ) in Ω in each of the subsequent cases: m,xn is a sequence in F V(Ω) ′ by Lemma 3.5 a). Moreover, we have T K m,xn (f ) = T K m (f )(x n ) for every f ∈ F V(Ω) which implies that (T K m,xn (f )) is a Cauchy sequence in K because T K m (f ) ∈ CC(Ω) by assumption. Since K is complete, it has a unique limit T (f ) ∶= lim n→∞ T K m,xn (f ) defining a linear functional in f . (i) The set N ∶= {x n n ∈ N} is precompact in Ω since Cauchy sequences are precompact. Hence there are j ∈ J, l ∈ L and C > 0 such that for every f ∈ F V(Ω). Therefore the set {T K m,xn n ∈ N} is equicontinuous in F V(Ω) ′ which implies that T ∈ F V(Ω) ′ and the convergence of (T K m,xn ) to T in F V(Ω) ′ γ due to the observation in the beginning and the fact that γ(F V(Ω) ′ , F V(Ω)) and σ(F V(Ω) ′ , F V(Ω)) coincide on equicontinuous sets. In particular, (T K m,xn ) is a Cauchy sequence in F V(Ω) ′ γ . (ii) There exist j ∈ J, l ∈ L and C > 0 such that for every f ∈ F V(Ω). Therefore the set {T K m,xn n ∈ N} is equicontinuous in F V(Ω) ′ and we conclude that T K m,xn is a Cauchy sequence in F V(Ω) ′ γ like in (i).
The subsequent proposition and lemma handle the same question as before but now for uniform continuity.

Proposition (uniform continuity). (T E
(z,x) , T K (z,x) ) (z,x)∈Muc is a strong and consistent subfamily for Proof. Let u ∈ F V(Ω)εE and (z, x) ∈ M uc . If (δ zn − δ xn ) converges to 0 in GV(Ω) ′ κ , then (S(u)(z n ) − S(u)(x n )) converges to 0 in E since u is uniformly continuous and u(δ zn − δ xn ) = S(u)(z n ) − S(u)(x n ). Hence we conclude S(u) ∈ ker T E (z,x) and We denote by UC(Ω) the space of scalar-valued uniformly continuous functions on a metric space Ω. We mean by BUC(Ω) the space of scalar-valued bounded, uniformly continuous functions equipped with the topology of uniform convergence on Ω.
There exist j ∈ J, l ∈ L and C > 0 such that sup for every f ∈ F V(Ω). Therefore the set {T K m,zn − T K m,xn n ∈ N} is equicontinuous in F V(Ω) ′ and we conclude the statement like before.

Proposition (continuous extendability). (T E ext , T K ext ) is a strong and consistent subfamily for
Proof. Let u ∈ F V(Ω)εE. From Proposition 4.3 we deduce S(u) ∈ C(Ω, E). Let x ∈ ∂Ω and (x n ) be a sequence in Ω converging to x. Then we have and thus in F V(Ω) ′ κ by the Banach-Steinhaus theorem. Hence we conclude 4.11. Lemma. Let F V(Ω) be a dom-space, Ω ⊂ X, X a metric space and As a consequence of the Banach-Steinhaus theorem we get δ x ○ (T K ext ○ T K m ) ∈ F V(Ω) ′ and the convergence in F V(Ω) ′ γ . Let us turn to differentiability.

4.12.
Proposition (differentiability on a subset). (T E ∂v , T K ∂v ) is a strong and consistent subfamily for Proof. Let u ∈ F V(Ω)εE and x ∈ ω. Then x ∈ ω as h → 0. The Banach-Steinhaus theorem yields the statement.
Our last proposition of this section is immediate. 4.14. Proposition (vanishing on a subset, additivity, homogeneity). (T E ω0 , T K ω0 ), (T E a , T K a ) and (T E h , T K h ) are strong and consistent subfamilies for (F V, E).

Examples
In our last section we treat many examples of spaces F V(Ω, E) of weighted functions 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 F V(Ω, E) ≅ F V(Ω)εE holds. For this purpose we recapitulate some definitions which are connected to different types of completeness of E. Let us recall the following definition from [41, 9-2-8 [40] and [8]. Furthermore, 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 also strict. The first two by [22, p. 58 We start with the simplest example of all. Let Ω be a non-empty set and equip the space E Ω with the topology of pointwise convergence, i.e. the locally convex topology given by the seminorms for finite K ⊂ Ω and α ∈ A. To prove E N0 ≅ K N0 εE for complete E is given as an exercise in [23, Aufgabe 10.5, p. 259] which we generalise now.

Example.
Let Ω be a non-empty set and E an lcHs. Then E Ω ≅ K Ω εE.
Proof. The strength and consistency of the defining family (id E Ω , id K Ω ) is obvious. Let f ∈ E Ω , K ⊂ Ω be finite and set 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 [22,6.7.4 Proposition,p. 113] implying E Ω ⊂ E Ω κ by Lemma 3.11 b) and our statement by virtue of Theorem 3.14 with Property 3.13 d).
The space of càdlàg functions on a set Ω ⊂ R with values in an lcHs E is defined by

Proposition.
Let Ω ⊂ R, K ⊂ Ω be compact and E an lcHs. Then f (K) is precompact for every f ∈ D(Ω, E).
Proof. Let f ∈ D(Ω, E), α ∈ A and ε > 0. We set f x ∶= lim w↗x f (w), for every x ∈ Ω, y ∈ E and r > 0. Let x ∈ Ω. Then there is r −x > 0 such that The sets V x are open in Ω with respect to the topology induced by R and K ⊂ ⋃ x∈K V x . Since K is compact, there are n ∈ N and x 1 , . . . , x n ∈ K such that K ⊂ ⋃ n i=1 V xi . Hence we get which means that f (K) is precompact.
Due to the preceding proposition the maps given by for compact K ⊂ Ω and α ∈ A form a system of seminorms inducing a locally convex topology on D(Ω, E). Further, D(Ω, E) = dom T E rc ∩ dom T E ll where the right-continuity is described by , and having limits from the left is described by Let Ω ⊂ R be locally compact and E an lcHs. If E is quasi-complete, then D(Ω)εE ≅ D(Ω, E).
Proof. First, we show that the defining family (T E m , T K m ) m∈{rc}∪{ll} for (D, E) is strong and consistent. The strength is a consequence of a simple calculation, so we only prove the consistency explicitely. Let x ∈ Ω, (x n ) be a sequence in Ω such that x n ↘ x resp. x n ↗ x. We have Since Ω is locally compact, there are a compact neighbourhood U (x) ⊂ Ω of x and n 0 ∈ N such that x n ∈ U (x) for all n ≥ n 0 . Hence we deduce for every f ∈ D(Ω). Therefore the set {δ xn n ≥ n 0 } is equicontinuous in D(Ω) ′ which implies that (δ xn ) converges to x n ↗ x, for every u ∈ D(Ω)εE follows the consistency. Second, let f ∈ D(Ω, E), K ⊂ Ω be compact and set N K (f ) ∶= f (Ω)χ K (Ω). We observe that Thus we deduce that N K (f ) is precompact in E for every f ∈ D(Ω, E) and every compact K ⊂ Ω by Proposition 5.2 and we obtain D(Ω, E) ⊂ D(Ω, E) κ by virtue of Lemma 3.11 b). The quasi-completeness of E yields that N K (f ) is relatively compact by [22,3.5.3 Proposition,p. 65] and that acx(N K (f )) is absolutely convex and compact. We derive our statement from Theorem 3.14 with Property 3.13 d).
Let us consider one of the most classical examples next, namely, the space C(Ω, E) of continuous functions on a k R -space Ω with values in an lcHs E equipped with the topology of uniform convergence on compact subsets of Ω, i.e. we choose the family of weights W given by ν K ∶= χ K for compact K ⊂ Ω. In [4, 2.4 Theorem (2), p. 138-139] Bierstedt proved that CW(Ω, E) ≅ CW(Ω)εE if E is quasi-complete which we improve now.
Proof. First, we observe that the defining family (T E c , T K c ) for (CW, E) is strong and consistent by Proposition 4.3 and Lemma 4.4 b)(i). Let f ∈ CW(Ω, E), K ⊂ Ω be compact and set If Ω is even metrisable, then f (K) is also metrisable by [9, Chap. IX, §2.10, Proposition 17, p. 159] and thus the finite union N K (f ) as well by [38, Theorem 1, p. 361] since the compact set N K (f ) is collectionwise normal and locally countably compact by [13,5.1.18 Theorem,p. 305]. Further, acx(N K (f )) is absolutely convex and compact in E if E has ccp resp. if Ω is metrisable and E has metric ccp. Thus we deduce CW(Ω, E) ⊂ CW(Ω, E) κ by Lemma 3.11 b). We conclude that CW(Ω, E) ≅ CW(Ω)εE if E has ccp resp. if Ω is metrisable and E has metric ccp by Theorem 3.14 with Property 3.13 d).
We proceed to spaces of distributions. Let us denote by D(U ) the linear subspace of the space C ∞ (U, K) of smooth functions consisting of all functions with compact support in an open subset U ⊂ R d which is equipped with its usual inductive limit topology. A distribution f ∈ L(D(U ), E) with an lcHs E and U = R d or where ϕ t (x) ∶= t d ϕ(tx) for x ∈ U and ⟨⋅, ⋅⟩ denotes the canonical pairing (see [19, Definition 3.2.2, p. 74]). By L λ-h (D(U ), E) we mean the space of all distributions which are homogeneous of degree λ and set D ′ (U ) λ-h ∶= L λ-h (D(U ), K). It is easily seen that g ∈ L(D(U ), E) is homogeneous of degree λ ∈ C if and only if g is in the kernel of the linear operator Let Ω be a metric space and E an lcHs. If E is a Fréchet or a semi-Montel space, then CC(Ω, E) ≅ CC(Ω)εE.
Proof. The defining family (T E m , T K m ) m∈{id}∪{cc} for (CC, E) is strong and consistent by Proposition 4.6 with Lemma 4.7 (i) for Cauchy continuity. First, we consider the case that E is a Fréchet space. Let f ∈ CC(Ω, E), K ⊂ Ω be precompact and set [1,Proposition 4.11,p. 576]. Thus we obtain CC(Ω, E) ⊂ CC(Ω, E) κ by virtue of Lemma 3.11 b). Since E is complete, the first part of the statement follows from Theorem 3.14 with Property 3.13 a). If E is a semi-Montel space, then it is a consequence of Corollary 3.16.
Let (Ω, d) be a metric space, E an lcHs and BUC(Ω, E) ∶= {f ∈ E Ω f uniformly continuous and bounded} be equipped with the system of seminorms given by for α ∈ A and let T E id ∶= id E Ω . 5.8. Example. Let (Ω, d) be a metric space and E an lcHs. If E is a semi-Montel space, then BUC(Ω, E) ≅ BUC(Ω)εE.
Proof. The defining family (T E m , T K m ) m∈{id}∪Muc for (BUC, E) is strong and consistent by Proposition 4.8 with Lemma 4.9 for uniform continuity yielding our statement by Corollary 3.16.
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

K. KRUSE
The topological subspace C [γ] z,0 (Ω, E) of γ-Hölder continuous functions that vanish at infinity consists of all f ∈ C for every w ∈ Ω. Then we have for every α ∈ A that z (Ω, E).

5.9.
Example. Let (Ω, d) be a metric space, z ∈ Ω, E be an lcHs and 0 < γ ≤ 1. Then z,0 (Ω)εE if Ω is precompact and E quasi-complete. Proof. Let us start with a). From Proposition 4.14 for vanishing at z and a simple calculation follows that (T E m , T K m ) m∈Ω∪{{z}0} is a strong and consistent family for (C z,0 (Ω, E) and 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 [35,Chap. I,5.1,p. 25]. Therefore N π⊂K δ ,1 (f ) is precompact for every δ > 0 connoting the precompactness of ∈ Ω × Ω} by the proof of Lemma 3.11 d). It follows that N 1 (f ) is relatively compact by [22,3.5.3 Proposition,p. 65] and K ∶= acx(N 1 (f )) is absolutely convex and compact if E is quasi-complete and thus has ccp. Hence statement b) is a consequence of Lemma 3.11 d) and Theorem 3.14 with Property 3.13 d). Now, we consider spaces of weighted continuously partially differentiable functions and present the counterpart for differentiable functions to Bierstedt's results [4,2.4 Theorem, and [4,2.12 Satz,p. 141] for the space CV(Ω, E) of continuous functions from a completely regular Hausdorff space Ω to an lcHs E equipped with a weighted topology given by a Nachbin-family V of weights and its topological subspace CV 0 (Ω, E) of functions which vanish at infinity in the weighted topology. We recall the following. A function f ∶ Ω → E on an open set Ω ⊂ R d to an lcHs E is called continuously partially differentiable (f is C 1 ) if for the n-th unit vector e n ∈ R d the limit if β n ≠ 0 and the right-hand side exists in E for every x ∈ Ω. Further, we define . Then we clearly have . . , k} if k < ∞ and ⟨k⟩ ∶= N 0 if k = ∞ and let V k ∶= ((ν j,l,β ) β∈N d 0 , β ≤l ) j∈J,l∈⟨k⟩ be a family of weights on an open set Ω ⊂ R d which is directed, i.e. ∀ j 1 , j 2 ∈ J, l 1 , l 2 ∈ ⟨k⟩ ∃ j 3 ∈ J, l 3 ∈ ⟨k⟩, l 3 ≥ max(l 1 , l 2 ), C > 0 ∀ i ∈ {1, 2}, β ≤ l i ∶ ν ji,li,β ≤ Cν j3,l3,β as well as ∀ x ∈ Ω, l ∈ ⟨k⟩ ∃ j ∈ J ∀ β ∈ N d 0 , β ≤ l ∶ 0 < ν j,l,β (x) (see Definition 3.1 and Remark 3.4). For k ∈ N ∞ and a directed family V k ∶= ((ν j,l,β ) β∈N d 0 , β ≤l ) j∈J,l∈⟨k⟩ of weights on an open set Ω ⊂ R d we define the space of weighted k-times continuously partially differentiable functions with values in an lcHs E as We define the topological subspace of CV k (Ω, E) consisting of the functions that vanish with all their derivatives when weighted at infinity by The following property for a family of directed weights allows us to use Lemma 4.4 (i). A directed family of weights V k is called locally bounded away from zero on 5.10. Example. Let E be an lcHs, k ∈ N ∞ , V k be a family of weights which is locally bounded away from zero on an open set Ω ⊂ R d .
∈M is a strong and consistent family for (CV k , E) resp. subfamily for We start with the proof of part a). To prove strength is quite simple and thus we concentrate on consistency which we prove by induction. Let σ ∈ Sym d . For follows from Proposition 4.3 and Lemma 4.4 (i) for continuity since V k is locally bounded away from zero.
Proof. We already know that the defining family is strong and consistent by Example 5.10 a). Let f ∈ CW k (Ω, E), K ⊂ Ω be compact, M l ∶= {β ∈ N d 0 β ≤ l} for l ∈ ⟨k⟩ and set N K,l (f ) is compact since it is a finite union of compact sets. Furthermore, the compact sets {0} and (∂ β ) E f (K) are metrisable by [9, Chap. IX, §2.10, Proposition 17, p. 159] and thus their finite union N K,l (f ) is metrisable as well by [38,Theorem 1,p. 361] since the compact set N K,l (f ) is collectionwise normal and locally countably compact by [13,5.1.18 Theorem,p. 305]. Due to Lemma 3.11 b) we obtain CW k (Ω, E) ⊂ CW k (Ω, E) κ for any lcHs E. If E has metric ccp, then the set acx(N K,l (f )) is absolutely convex and compact. Thus Theorem 3.14 with Property 3.13 d) settles the case for k < ∞. If k = ∞ and E is locally complete, we observe that K β ∶= acx((∂ β ) E f (K)) for f ∈ CW ∞ (Ω, E) is absolutely convex and compact by [8,Proposition 2,p. 354]. Then we have N K,l (f ) ⊂ acx ⋃ β ≤l K β and the set on the right-hand side is absolutely convex and compact by [22,6.7.3 Proposition,p. 113]. Again, the statement follows from Theorem 3.14 with Property 3.13 d).
In the context of differentiability on infinite dimensional spaces the preceding example remains true for an open subset Ω of a Fréchet space or DFM-space and quasi-complete E by [32,3.2 Corollary,p. 286]. Like here this can be generalised to E with [metric] ccp. Let us consider kernels of linear partial differential operators next. Let E be an lcHs, k ∈ N ∞ and Ω ⊂ R d open. Let n ∈ N, β m ∈ N d 0 with β m ∈ ⟨k⟩ and a m ∶ Ω → K for 1 ≤ m ≤ n. We set dom P (∂) E ∶= ⋂ n m=1 dom(∂ βm ) E c and For a directed family of weights V k on Ω we define the topological subspaces of CV k (Ω, E) given by CV k P (∂) (Ω, E) ∶= {f ∈ CV k (Ω, E) f ∈ ker P (∂) E } and CV k 0,P (∂) (Ω, E) ∶= {f ∈ CV k 0 (Ω, E) f ∈ ker P (∂) E }.

5.13.
Example. Let E be an lcHs, k ∈ N ∞ , V k be a family of weights which is locally bounded away from zero on an open set Ω ⊂ R d .
A special case of example d) is already known to be a consequence of [7, Theorem 9, p. 232], namely, if k = ∞ and P (∂) is hypoelliptic with constant coefficients. In particular, this covers the space of holomorphic functions and the space of harmonic functions. The special case of example b) of holomorphic functions with exponential growth on strips is handled in [28,3.11 Theorem,p. 31]. Holomorphy on infinite dimensional spaces is treated in [11,Corollary 6.35, where V = W, Ω is an open subset of a locally 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. 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 ⊂ R d open and bounded. The space C k (Ω, E) is given by C k (Ω, E) ∶= {f ∈ C k (Ω, E) (∂ β ) E f cont. extendable on Ω for all β ∈ N d 0 } and equipped with the system of seminorms given by for α ∈ A if k < ∞ and by 14. Example. Let E be an lcHs, k ∈ N ∞ and Ω ⊂ R d open and bounded. Then C k (Ω, E) ≅ C k (Ω)εE if E has metric ccp.
Proof. The defining family is the union of the families (σ(∂ β ) E c , σ(∂ β ) K c ) (σ,β)∈M with M ∶= Sym d ×{γ ∈ N d 0 γ ≤ k} for continuous partial differentiability on Ω and ((∂ β ) E ext , (∂ β ) K ext ) β ∈⟨k⟩ for continuous extendability of the partial derivatives on ∂Ω. The strength and consistency of the first subfamily for continuous partial differentiability follows like in Example 5.10 since C k (Ω) is a Banach space if k < ∞ and a Fréchet space if k = ∞, in particular, both are barrelled. The strength of the second subfamily for continuous extendability of the partial derivatives is clear and its consistency follows from Proposition 4.10, Lemma 4.11 and S(u) ∈ dom(∂ β ) E c for all u ∈ C k (Ω)εE and β ∈ ⟨k⟩ by the consistency of the first subfamily. Let f ∈ C k (Ω, E), l ∈ N 0 and set We denote by f β the continuous extension of (∂ β ) 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 β (Ω) like in Example 5.12. Due to Lemma 3.11 b) and Theorem 3.14 with Property 3.13 d) we obtain our statement if E has metric ccp.
We close our last section with spaces of ultradifferentiable functions. Let E be an lcHs, Ω ⊂ R d open, K ∶= {K ⊂ Ω K compact} and (M p ) p∈N0 be a sequence of positive numbers. The space E (Mp) (Ω, E) of ultradifferentiable functions of class (M p ) of Beurling-type is defined as