Extension of vector-valued functions and weak-strong principles for differentiable functions of finite order

In this paper we study the problem of extending functions with values in a locally convex Hausdorff space $E$ over a field $\mathbb{K}$, which have weak extensions in a weighted Banach space $\mathcal{F}\nu(\Omega,\mathbb{K})$ of scalar-valued functions on a set $\Omega$, to functions in a vector-valued counterpart $\mathcal{F}\nu(\Omega,E)$ of $\mathcal{F}\nu(\Omega,\mathbb{K})$. Our findings rely on a description of vector-valued functions as linear continuous operators and extend results of Frerick, Jord\'{a} and Wengenroth. As an application we derive weak-strong principles for continuously partially differentiable functions of finite order, vector-valued versions of Blaschke's convergence theorem for several spaces and Wolff type descriptions of dual spaces.

Often, the underlying idea to prove such an extension theorem is to use a representation of an E-valued function by a continuous linear operator. Namely, if F (Ω) ∶= F (Ω, K) is a locally convex Hausdorff space of scalar-valued functions on a set Ω such that the point evaluations δ x at x belong to the dual F (Ω) ′ for each x ∈ Ω, then the function S(u)∶ Ω → E given by x ↦ u(δ x ) is well-defined for every element u of Schwartz' ε-product F (Ω)εE ∶= L e (F (Ω) ′ κ , E) where the dual F (Ω) ′ is equipped with the topology of uniform convergence on absolutely convex compacts subsets of F (Ω), the space of continuous linear operators L(F (Ω) ′ κ , E) is equipped with the topology of uniform convergence on the equicontinuous subsets of F (Ω) ′ κ and E is a locally convex Hausdorff space over the field K. In many cases the function S(u) inherits properties of the functions in F (Ω), e.g. if F (Ω) = (O(Ω), τ co ) is the space of holomorphic functions on an open set Ω ⊂ C equipped with the compact-open topology τ co , then the space of functions of the form S(u) with u ∈ (O(Ω), τ co )εE coincides with the space O(Ω, E) of E-valued holomorphic functions if E is locally complete. Even more is true, namely, that the map S∶ (O(Ω), τ co )εE → (O(Ω, E), τ co ) is a (topological) isomorphism (see [8, p. 232]). So suppose that there is a locally convex Hausdorff space F (Ω, E) of Evalued functions on Ω such that the map S∶ F (Ω)εE → F (Ω, E) is well-defined and at least a (topological) isomorphism into, i.e. to its range. The precise formulation of the extension problem from the beginning is the following question.
1.1. 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 ? Even the case Λ = Ω is interesting because then the question is about properties of vector-valued functions and a positive answer is usually called a weak-strong principle. From the connection of F (Ω)εE and F (Ω, E) it is evident to seek for extension theorems for vector-valued functions by extension theorems for continuous linear operators. In this way many of the extension theorems of the aforementioned references are derived but in most of the cases the space F (Ω) has to be a semi-Montel (see [31,36,46]) or even a Fréchet-Schwartz space (see [8,25,31,33,34,35,40,46]) or E is restricted to be a semi-Montel space (see [6,46]). The restriction to semi-Montel spaces F (Ω) resp. E, i.e. to locally convex spaces in which every bounded set is relatively compact, is quite natural due to the topology of the dual F (Ω) ′ κ in the ε-product F (Ω)εE and its symmetry F (Ω)εE ≅ EεF (Ω). In the present paper we treat the case of a Banach space which we denote by F ν(Ω) because its topology is induced by a weight ν. We use the methods developped in [26] and [41] where, in particular, the special case that F ν(Ω) is the space of bounded smooth functions on an open set Ω ⊂ R d 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 the infinite dimensional Banach space F ν(Ω) is compensated in [26] and [41] by using an auxiliary locally convex Hausdorff space F (Ω) of scalar-valued functions on Ω such that the closed unit ball of F ν(Ω) is compact in F (Ω). They share the property that S∶ F ν(Ω)εE → F ν(Ω, E) and S∶ F (Ω)εE → F (Ω, E) are topological isomorphisms into but usually it is only known in the latter case that S is surjective as well under some mild completeness assumption on E. For instance, if F ν(Ω, E) ∶= H ∞ (Ω, E) is the space of bounded holomorphic functions on an open set Ω ⊂ C with values in a locally complete space E, then the space F (Ω, E) ∶= (O(Ω, E), τ co ) is used in [26].
Let us outline the content of our paper. We give a general approach to the extension problem for Banach function spaces F ν(Ω). It combines the methods of [26,41] with the ones of [44] as in [46] which require that the spaces F ν(Ω) and F ν(Ω, E) have a certain structure (see Definition 2.3). To answer Question 1.1 we have to balance the sets Λ ⊂ Ω and the spaces G ⊂ E ′ . If we choose Λ to be 'thin', then G has to be 'thick' (see Section 3 and 5) and vice versa (see Section 4). In Section 6 we use the results of Section 3 to derive and improve weak-strong principles for differentiable functions of finite order. Section 7 is devoted to vectorvalued Blaschke theorems and Section 8 to Wolff type descriptions of the dual of F (Ω).

Notation and Preliminaries
We use essentially the same notation and preliminaries as in [46,Section 2]. We equip the spaces R d , d ∈ N, and C with the usual Euclidean norm ⋅ . 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 more details on the theory of locally convex spaces see [24,38,52].
By X Ω we denote the set of maps from a non-empty set Ω to a non-empty set 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 . We write F ≅ E if F and E are (linearly topologically) isomorphic. 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 and on the bounded subsets of F if t = b. We use the symbol t(F ′ , F ) for the corresponding topology on F ′ . A linear subspace G of F ′ is called separating if f ′ (x) = 0 for every f ′ ∈ G implies x = 0. This is equivalent to G being σ(F ′ , F )-dense (and κ(F ′ , F )dense) in F ′ by the bipolar theorem. 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 [38, p. 151]). The space F is called locally complete if F D is a Banach space for every closed disk D ⊂ F (see [38,10.2.1 Proposition,p. 197]).
Furthermore, we recall the definition of continuous partial differentiability of a vector-valued function that we need in many examples, especially, for the weakstrong principle for differentiable functions of finite order in Section 6. 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 E = K, we usually write ∂ β f ∶= (∂ β ) K f . We denote by τ C k the usual topology on C k (Ω, E), namely, the locally convex topology given by the seminorms In addition, we use the following notion for the relation between the ε-product F (Ω)εE and the space F (Ω, E) of vector-valued functions that has already been described in the introduction.
Let Ω be a nonempty set and E an lcHs. Let F (Ω) ⊂ K Ω and F (Ω, E) ⊂ E Ω be lcHs such that δ x ∈ F (Ω) ′ for all x ∈ Ω. We call the spaces F (Ω) and F (Ω, , is a well-defined isomorphism into. We call F (Ω) and F (Ω, E) ε-compatible if S is an isomorphism. If we want to emphasise the dependency on F (Ω), we write S F (Ω) instead of S.

2.2.
Definition (strong, consistent). Let Ω and ω be non-empty sets and E, F (Ω) ⊂ K Ω and F (Ω, E) ⊂ E Ω be lcHs. Let δ x ∈ F (Ω) ′ for all x ∈ Ω and T K ∶ F (Ω) → K ω and T E ∶ F (Ω, E) → E ω be linear maps. a) We call (T E , T K ) a consistent family for (F , E) if we have for every u ∈ F (Ω)εE that S(u) ∈ F (Ω, E) and This is a special case of [46,2.2 Definition,p. 4] where the considered family (T E m , T K m ) m∈M only consists of one pair, i.e. the set M is a singleton. In the introduction we have already hinted that the spaces F (Ω) and F (Ω, E) for which we want to prove extension theorems need to have a certain structure, namely, the following one.

Definition (generator).
Let Ω and ω be non-empty sets, ν∶ ω → (0, ∞), This is a special case of [44, Definition 3, p. 1515] where the family of weights only consists of one weight function. For instance, if Ω ∶= ω, T E ∶= id E Ω and ν ∶= 1 on Ω, then F ν(Ω, E) is the linear subspace of F (Ω, E) consisting of bounded functions, in particular, if Ω ⊂ C is open and F (Ω, E) ∶= O(Ω, E), then F ν(Ω, E) = H ∞ (Ω, E) is the space of E-valued bounded holomorphic functions on Ω. Due to (E, (p α ) α∈A ) being an lcHs with directed system of seminorms the topology of F ν(Ω, E) generated by ( ⋅ α ) α∈A is locally convex and the system ( ⋅ α ) α∈A is directed but need not be Hausdorff.
Proof. Part a) follows from the continuity of i and the ε-into-compatibility of F (Ω) and F (Ω, E). Let us turn to part b). As in [44, Lemma 7, p. 1517] it follows from the bipolar theorem that x ∈ ω} on the right-hand side, and that sup by consistency, which proves part b). Let us address part c). The continuity of i implies the continuity of the inclusion F ν(Ω)εE ↪ F (Ω)εE and thus we obtain u F (Ω) ′ ∈ F (Ω)εE for every u ∈ F ν(Ω)εE. If u ∈ F ν(Ω)εE and α ∈ A, then there are C 0 , C 1 > 0 and an absolutely convex Let us turn to part d). We have u F (Ω) ′ ∈ F (Ω)εE for every u ∈ F ν(Ω)εE and 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 (F ν, E) by [ x ∈ ω. It follows that e ′ ○ f ∈ F ν(Ω) for all e ′ ∈ E ′ and so that (T E , T K ) is a strong generator for (F ν, E). Thus part (i) holds and implies part (ii) by the first part of the proof of [44, Theorem 14, p. 1524].
The canonical situation in part c) is that F ε ν(Ω, E) and F ν(Ω, E) coincide as linear spaces for locally complete E as we will encounter in the forthcoming examples, e.g. if F ν(Ω, E) ∶= H ∞ (Ω, E) and F (Ω, E) ∶= (O(Ω, E), τ co ) for an open set Ω ⊂ C. That all three spaces in part c) coincide is usually only guaranteed by [44,Theorem 14 (ii), p. 1524] if E is a semi-Montel space. Therefore the 'mingle-mangle' space F ε ν(Ω, E) is a good replacement for S(F ν(Ω)εE) for our purpose.

Extension of vector-valued functions
In this section the sets from which we want to extend our functions are 'thin'. They are so-called sets of uniqueness.
3.1. Definition (set of uniqueness). Let F ν(Ω) be Hausdorff. A set U ⊂ ω is called a set of uniqeness for x ∈ U } is weak *dense in F ν(Ω) ′ by the bipolar theorem if U is a set of uniqueness for (T K , F ν). The set U ∶= ω is always a set of uniqueness for (T K , F ν) as F ν(Ω) is an lcHs by assumption. Next, we introduce the notion of a restriction space which is a special case of [46, 3.3 3.2. Definition (restriction space). Let G ⊂ E ′ be a separating subspace and U a set of uniqueness for The time has come to use our auxiliary spaces F (Ω), F (Ω, E) and F ε ν(Ω, E) from Proposition 2.4.

3.3.
Remark. Let (T E , T K ) be a strong, consistent family for (F , E) and a generator for (F ν, E). Let F (Ω) and F (Ω, E) be ε-into-compatible and the inclusion F ν(Ω) ↪ F (Ω) continuous. Consider a set of uniqueness U for (T K , F ν) and a for all x ∈ U and f e ′ ∶= e ′ ○ f ∈ F (Ω) for each e ′ ∈ E ′ by the strength of the family.
for every x ∈ ω, which implies that for every e ′ ∈ E ′ there are α ∈ A and C > 0 such that p α (u(y ′ )) < ∞ by strength and consistency. Hence f e ′ ∈ F ν(Ω) for every e ′ ∈ E ′ andf ∈ F ν G (U, E).
Under the assumptions of Remark 3.3 the map is well-defined and linear. In addition, we derive from (1) that R U,G is injective since U is a set of uniqueness and G ⊂ E ′ separating.
3.4. Question. Let the assumptions of Remark 3.3 be fulfilled. When is the injective restriction map surjective?
Due to Proposition 2.4 c) the Question 1.1 is a special case of this question if Λ ⊂ Ω =∶ ω and U ∶= Λ is a set of uniqueness for (id K Ω , F ν). To answer Question 3.4 for general sets of uniqueness we have to restrict to a certain class of 'thick' separating subspaces of E ′ . . Let E be locally complete, G ⊂ E ′ determine boundedness, Z a Banach space whose closed unit ball B Z 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 3.7. Theorem. Let E be locally complete, G ⊂ E ′ determine boundedness and F (Ω) and F (Ω, E) be ε-into-compatible. Let (T E , T K ) be a generator for (F ν, E) and a strong, consistent family for (F , E), F ν(Ω) a Banach space whose closed unit ball B F ν(Ω) is a compact subset of F (Ω) and U a set of uniqueness for (T K , F ν). Then the restriction map

Definition
is surjective. Proof for every e ′ ∈ G and x ∈ U we have that A is σ(X, Z)-σ(E, G)-continuous. We apply Proposition 3.6 and gain an extensionÂ ∈ Y εE of A such thatÂ(B ○Y ′ Z ) is bounded in E. We set F ∶= S(Â) ∈ F ε ν(Ω, E) and get for all x ∈ U that Let Ω ⊂ R d 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 define the space (Ω, E) f ∈ ker P (∂) E } of zero solutions and for a continuous weight ν∶ Ω → (0, ∞) the weighted space of zero solutions

Proposition.
Let Ω ⊂ R d be open, E a locally complete lcHs and P (∂) K a hypoelliptic linear partial differential operator. Then via S (C ∞ P (∂) (Ω),τco) holds Proof. We already know that is an isomorphism by [44, Example 18 b), p. 1528]. From τ co = τ C ∞ on C ∞ P (∂) (Ω) by the hypoellipticity of P (∂) K (see e.g. [26, p. 690 In particular, we obtain that is an isomorphism. From the first part of the proof of [44,Theorem 14,p as locally convex spaces, which proves our statement. yielding Further, there is N ε ∈ N such that for all n, m ≥ N ε it holds that Hence for n ≥ N ε we choose m ≥ max(N ε , N ε,x ) and derive It follows that f n −f ν ≤ ε and f ν ≤ ε+ f n ν for all n ≥ N ε , implying the convergence of (f n ) n∈N to f in Cν ∞ P (∂) (Ω).

3.10.
Corollary. Let E be a locally complete lcHs, G ⊂ E ′ determine boundedness, . We note that F (Ω) and F (Ω, E) are εcompatible and (T E , T K ) is a strong, consistent family for (F , E) by Proposition 3.8. We observe that F ν(Ω) is a Banach space by Proposition 3.9 and for every compact K ⊂ Ω we have is a Fréchet-Schwartz space, thus a Montel space, and it is easy to check that B F ν(Ω) is τ co -closed. Hence the bounded and τ co -closed set B F ν(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 (id E Ω , id K Ω ) for (F , E) gives F ε ν(Ω, E) = F ν(Ω, E) as linear spaces by Proposition 2.4 c). From Theorem 3.7 follows our statement.
If Ω = D ⊂ C is the open unit disc, P (∂) = ∂ the Cauchy-Riemann operator and For a continuous function ν∶ D → (0, ∞) and a complex lcHs E we define the Bloch type spaces and the complex derivative 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), τ co ) analogously to the proof of Proposition 3.9.
Let E be an lcHs and ν∶ D → (0, ∞) be continuous. We Then we have for every α ∈ A that  (4) by the identity theorem, proving our statement by Theorem 3.7.

Extension of locally bounded functions
In order to obtain an affirmative answer to Question 3.4 for general separating subspaces of E ′ we have to restrict to a certain class of 'thick' sets of uniqueness.
In particular, U is a set of uniqueness if it fixes the topology. The present definition of fixing the topology is a special case of [46,4.1 Definition,p. 18]. Sets that fix the topolgy appear under many different names, e.g. dominating, (weakly) sufficient, sampling sets (see [46, p. 18-19] and the references therein), and they are related to ℓν(U )-frames used by Bonet et. al in [9]. For a set U , a function ν∶ U → (0, ∞) and an lcHs E we set If U is countable and fixes the topology in F ν(Ω), the inclusion ℓν(U ) ↪ (K U , τ co ) is continuous and ℓν(U ) contains the space of sequences (on U ) with compact support as a linear subspace, then (T K x ) x∈U is an ℓν(U )-frame in the sense of [

4.2.
Definition (lb-restriction space). Let F ν(Ω) be a Hausdorff space, U fix the topology in F ν(Ω) and G ⊂ E ′ a separating subspace. We set Let us recall the assumptions of Remark 3.3 but now U fixes the topology. Let (T E , T K ) be a strong, consistent family for (F , E) and a generator for (F ν, E).
Let F (Ω) and F (Ω, E) be ε-into-compatible and the inclusion F ν(Ω) ↪ F (Ω) continuous. Consider a set U which fixes the topology in F ν(Ω) and a separating subspace is well-defined and the question we want to answer is: Question. Let the assumptions of Remark 3.3 be fulfilled and U fix the topology in F ν(Ω). When is the injective restriction map . Let E be locally complete, G ⊂ E ′ a separating subspace and Z a Banach space whose closed unit ball B Z is a compact subset of an lcHs Y . Let The following theorem is a generalisation of [

4.5.
Theorem. Let E be locally complete, G ⊂ E ′ a separating subspace and F (Ω) and F (Ω, E) be ε-into-compatible. Let (T E , T K ) be a generator for (F ν, E) and a strong, consistent family for (F , E), F ν(Ω) a Banach space whose closed unit ball B F ν(Ω) is a compact subset of F (Ω) and U fix the topology in F ν(Ω). Then the restriction map 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 ). Again, this equation allows us to consider f e ′ as a linear form on , which yields e ′ ○A ∈ F ν(Ω) = Z for all e ′ ∈ G. Hence we can apply Proposition 4.4 and obtain an extensionÂ ∈ Y εE (Ω), our statement follows directly from Theorem 4.5 whose conditions are fulfilled by the proof of Corollary 3.10. If
Sets that fix the topology in Bν(D) play an important role in the characterisation of composition operators on Bν(D) with closed range. Chen and Gauthier give a characterisation in [16] for weights of the form ν(z) = (1 − z 2 ) α , z ∈ D, 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 and define the pseudohyperbolic distance (see [16, p. 195-196]). For 0 < r < 1 a set E ⊂ D is called a pseudo r-net if for every w ∈ D there is z ∈ D with ρ(z, w) ≤ r (see [16, p. 198] Then the following statements are equivalent.

Extension of sequentially bounded functions
In this section we restrict to the case that E is a Fréchet space.

Definition ([8, Definition 12, p. 8])
. Let E be a Fréchet space. An increasing sequence (B n ) n∈N of bounded subsets of E ′ b fixes the topology in E if (B ○ n ) n∈N is a fundamental system of zero neighbourhoods of E.
In particular, if E is a Banach space, then an almost norming set B ⊂ E ′ , i.e. B is bounded w.r.t. to the operator norm and the polar B ○ is bounded in E, fixes the topology in E and we refer the reader to [ (sb-restriction space). Let E be a Fréchet space, (B n ) fix the topology in E and G ∶= span(⋃ n∈N B n ). Let F ν(Ω) be a Hausdorff space, U a set of uniqueness for (T K , F ν) and set

Definition
Let E be a Fréchet space, (B n ) fix the topology in E and recall the assumptions of Remark 3.3. Let (T E , T K ) be a strong, consistent family for (F , E) and a generator for (F V, E). Let F (Ω) and F (Ω, E) be ε-into-compatible and the inclusion F ν(Ω) ↪ F (Ω) continuous. Consider a set of uniqueness U for (T K , F ν) and is well-defined.

Question.
Let the assumptions of Remark 3.3 be fulfilled, E be a Fréchet space, (B n ) fix the topology in E and G ∶= span(⋃ n∈N B n ). When is the injective restriction map

Corollary.
Let E be a Fréchet space, (B n ) fix the topology in E, set G ∶= span(⋃ n∈N B n ) and let F (Ω) and F (Ω, E) be ε-into-compatible. Let (T E , T K ) be a generator for (F ν, E) and a strong, consistent family for (F , E), F ν(Ω) a Banach space whose closed unit ball B F ν(Ω) is a compact subset of F (Ω) and U a set of uniqueness for (T K , F ν). Then the restriction map is endowed with its Banach space topology for n ∈ N, determines boundedness in E. Hence we conclude that f ∈ F ν E ′ ((Bn) n∈N ) (U, E), which yields that there is ) ⊂ E such that R U,G (S(u)) = f by Theorem 3.7.
As an application we directly obtain the following two corollaries of Corollary 5.4 since its assumptions are fulfilled by the proof of Corollary 3.10 and Corollary 3.12, respectively. 5.5. Corollary. Let E be a Fréchet space, (B n ) fix the topology in E and G ∶= span (⋃ n∈N B n ), Ω ⊂ R d open, P (∂) K a hypoelliptic linear partial differential operator, ν∶ Ω → (0, ∞) continuous and U a set of uniqueness for (id K Ω , Cν ∞ P (∂) ). If f ∶ U → E is a function such that e ′ ○ f admits an extension f e ′ ∈ Cν ∞ P (∂) (Ω) for each e ′ ∈ G and {f e ′ e ′ ∈ B n } is bounded in Cν ∞ P (∂) (Ω) for each n ∈ N, then there exists a unique extension F ∈ Cν ∞ P (∂) (Ω, E) of f . 5.6. Corollary. Let E be a Fréchet space, (B n ) fix the topology in E and G ∶= span(⋃ n∈N B n ), ν∶ D → (0, ∞) continuous and U * ⊂ D have an accumulation point (1, z)) for all z ∈ U * and {f e ′ e ′ ∈ B n } is bounded in Bν(D) for each n ∈ N, then there exists a unique F ∈ Bν(D, E) with

Weak-strong principles for differentiable functions of finite order
This section is dedicated to C k -weak-strong principles for differentiable functions. So the question is: An affirmative answer to the preceding question is called a C k -weak-strong principle. It is a result of Bierstedt [6, 2.10 Lemma, p. 140] that for k = 0 the C 0 -weakstrong principle holds if Ω ⊂ R d is open (or more general a k R -space), G = E ′ and E is such that every bounded set is already precompact in E. For instance, the last condition is fulfilled if E is a semi-Montel or Schwartz space. The C 0 -weak-strong principle does not hold for general E by [ set Ω ⊂ R d with values in a quasi-complete lcHs E is already C k , i.e. that from e ′ ○ f ∈ C k+1 (Ω) for all e ′ ∈ E ′ it follows f ∈ C k (Ω, E). A detailed proof of this statement is given by Schwartz in [61], simultaneously weakening the condition from quasi-completeness of E to sequential completeness and from weakly-C k+1 to weakly-C k,1 loc . 6.2. Theorem ([61, Appendice, Lemme II, Remarques 1 0 ), p. 146-147]). Let E be a sequentially complete lcHs, Here C k,1 loc (Ω) denotes the space of functions in C k (Ω) whose partial derivatives of order k are locally Lipschitz continuous. Part b) clearly implies a C ∞ -weak-  [43, 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 6.2. We start with the following definition. For k ∈ N 0 we define the space of k-times continuously partially differentiable E-valued functions on an open set Ω ⊂ R d whose partial derivatives up to order k are continuously extendable to the boundary of Ω by on Ω for all β ∈ N d 0 , β ≤ k} which we equip with the system of seminorms given by The space of functions in C k (Ω, E) such that all its k-th partial derivatives are γ-Hölder continuous with 0 < γ ≤ 1 is given by We set , (β, (x, y)) ∈ ω 2 . and the weight ν∶ ω → (0, ∞) by ν(β, x) ∶= 1, (β, x) ∈ ω 1 , and ν(β, (x, y)) ∶= 1 x − y γ , (β, (x, y)) ∈ ω 2 .
Proof. We take F (Ω) ∶= C k (Ω) and F (Ω, E) ∶= C k (Ω, E) and have F ν(Ω) = C k,γ (Ω) and F ν(Ω, E) = C k,γ (Ω, E) with the weight ν and generator (T E , T K ) for (F ν, E) described above. Due to the proof of [44, Example 20, p. 1529] and the first part of the proof of [44, Theorem 14, p. 1524] the spaces F (Ω) and F (Ω, E) are ε-intocompatible for any lcHs E (the condition that E has metric ccp in [44, Example 20, p. 1529] is only needed for ε-compatibility). Another consequence of [44, Example 20, p. 1529] is that holds for all u ∈ F (Ω)εE, implying Thus (T E , T K ) is a consistent family for (F , E) and its strength is easily seen. In addition, F ν(Ω) = C k,γ (Ω) is a Banach space by [ , whose proofs can be adjusted without additional assumptions (see Corollary 6.5 and Corollary 7.5 for this).
Next, we use the preceding corollary to generalise the theorem of Grothendieck and Schwartz on weakly C k+1 -functions. For k ∈ N 0 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 ⊂ R d by and Using Corollary 6.3, we are able to improve Theorem 6.2 to the following form.
Proof. Let us start with a). Let f ∶ Ω → E be such that e ′ ○ f ∈ C k,γ loc (Ω) for all e ′ ∈ G. Let (Ω n ) n∈N be an exhaustion of Ω with open, relatively compact sets Ω n ⊂ Ω with Lipschitz boundaries ∂Ω n (e.g. choose each Ω n as the interior of a finite union of closed axis-parallel cubes, see the proof of [68,Theorem 1.4,p. 7] for the construction) that satisfies Ω n ⊂ Ω n+1 for all n ∈ N. Then the restriction of e ′ ○ f to Ω n is an element of C k,γ (Ω n ) for every e ′ ∈ G and n ∈ N. Due to Corollary 6.3 we obtain that f ∈ C k,γ (Ω n , E) for every n ∈ N. Thus f ∈ C k,γ loc (Ω, E) since differentiability is a local property and for each compact K ⊂ Ω there is n ∈ N such that K ⊂ Ω n .
Let us turn to b), i.e. let f ∶ Ω → E be such that e ′ ○ f ∈ C k+1 (Ω) for all e ′ ∈ G.
by the mean value theorem applied to the real and imaginary part where It follows from part a) that f ∈ C k,1 loc (Ω, E). If Ω = R, γ = 1 and G = E ′ , then part a) of Corollary 6.5 is already known by [43,2.3 Corollary,p. 15]. A 'full' C k -weak-strong principle for k < ∞, i.e. the conditions of part b) imply f ∈ C k+1 (Ω, E), does not hold in general (see [43, p. 11-12]) but it holds if we restrict the class of admissible lcHs E. 6.6. Theorem. Let E be a semi-Montel space, G ⊂ E ′ determine boundedness, Ω ⊂ R d open and k ∈ N. If f ∶ Ω → E is such that e ′ ○ f ∈ C k (Ω) for all e ′ ∈ G, then f ∈ C k (Ω, E).
Proof. Let f ∶ Ω → E be such that e ′ ○ f ∈ C k (Ω) for all e ′ ∈ G. Due to Corollary 6.5 b) we already know that f ∈ C k−1,1 loc (Ω, E) since semi-Montel spaces are quasicomplete and thus locally complete. Now, let x ∈ Ω, ε x > 0 such that B(x, ε x ) ⊂ Ω, β ∈ N d 0 with β = k − 1 and n ∈ N with 1 ≤ n ≤ d. The set From the compactness of B we deduce that there is a subnet (h mι ) ι∈I , where I is a directed set, of (h m ) m∈N and y x ∈ B with Further, we note that the limit exists for all e ′ ∈ G and that (e ′ (y ι )) ι∈I is a subnet of the net of difference quotients on the right-hand side of (6) as ∂ β (e ′ ○ f ) = e ′ ○ (∂ β ) E f . Therefore for all e ′ ∈ G. By [46, 4.10 Proposition (i), p. 21] the topology σ(E, G) and the initial topology of E coincide on B. Combining this fact with (7), we deduce that In addition, for all e ′ ∈ G, meaning that the restriction of (∂ β+en ) E f on B(x, ε x ) to (E, σ(E, G)) is continuous, and the range (∂ β+en ) E f (B(x, ε x )) is bounded in E. As before we use that σ(E, G) and the initial topology of E coincide on (∂ β+en ) E f (B(x, ε x )), which implies that the restriction of (∂ β+en ) E f on B(x, ε 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 ) E f 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 [14,Lemma 4,p. 15]. This proof uses the Banach-Steinhaus theorem and needs the barrelledness of the Montel space E ′ b . Our weak-strong principle Theorem 6.6 does not need the barrelledness of E, hence can be applied to non-barrelled semi-Montel spaces like E = (C ∞ ∂,b (D), β) where β is the strict topology (see page 24, [46, 3.14 Proposition, p. 12] and [46,3.15 Remark,p. 13]).
Besides the 'full' C k -weak-strong principle for k < ∞ and semi-Montel E, part b) of Corollary 6.5 also suggests an 'almost' C k -weak-strong principle in terms of [23, 3.1.6 Rademacher's theorem, p. 216], which we prepare next. 6.7. 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 differentiable 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 [49,Definition 2.2,p. 17]. In particular, every real nuclear Fréchet lattice (see [32,Theorem 6,Corollary,p. 375,378]) and more general every real Fréchet-Montel space is a generalised Gelfand space by [49,Theorem 2.9,p. 18]. If E is a Banach space, then this definition coincides with the definition of a Gelfand space given in [19,Definition 4  6.8. 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 ∈ C k (Ω) for all e ′ ∈ G, then f ∈ C k−1,1 loc (Ω, E) and the derivative (∂ k ) E f (x) exists for Lebesgue almost all x ∈ Ω.
Proof. The first part follows from Corollary 6.5 b). Now, let [a, b] ⊂ Ω be a bounded interval. We set + x(b − a)). Then F is Lipschitz continuous as f ∈ C k−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 ) E f is differentiable on [a, b] almost everywhere. Since the open set Ω ⊂ R can be written as a countable union of disjoint open intervals I n , n ∈ N, and each I n is a countable union of closed bounded intervals [a m , b m ], 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. 6.9. Question.
(i) Are there other spaces than semi-Montel spaces E for which the 'full' C k -weak-strong principle Theorem 6.6 with k < ∞ is true? For instance, if k = 0, then it is still true if E is an lcHs such that every bounded set is already precompact in E by [6,2.10 Lemma,p. 140]. Does this hold for 0 < k < ∞ as well? (ii) Does the 'almost' C k -weak-strong principle Corollary 6.8 also hold for d > 1? bounded functions in the kernel of a hypoelliptic linear partial differential operator. Blaschke's convergence theorem says that if (z n ) n∈N ⊂ D is a sequence of distinct elements with ∑ n∈N (1 − z n ) = ∞ and if (f k ) k∈N is a bounded sequence in H ∞ (D) such that (f k (z n )) k converges in C for each n ∈ N, then there is f ∈ H ∞ (D) such that (f k ) k converges uniformly to f on the compact subsets of D, i.e. w.r.t. to τ co . 7.1. Proposition ([26, Proposition 4.1, p. 695]). Let (E, ⋅ ) be a Banach space, Z a Banach space whose closed unit ball B Z is a compact subset of an lcHs Y and let (A ι ) ι∈I be a net in Y εE such that Assume further that there exists a σ(Y ′ , Z)-dense subspace X ⊂ Y ′ such that lim ι A ι (x) exists for each x ∈ X. Then there is A ∈ Y εE with A(B ○Y ′ Z ) bounded and lim ι A ι = A uniformly on the equicontinuous subsets of Y ′ , i.e. for all equicontinuous B ⊂ Y ′ and ε > 0 there exists ς ∈ I such that Next, we generalise [26, Corollary 4.2, p. 695].
7.2. Corollary. Let (E, ⋅ ) be a Banach space and F (Ω) and F (Ω, E) be ε-intocompatible. Let (T E , T K ) be a generator for (F ν, E) and a strong, consistent family for (F , E), F ν(Ω) a Banach space whose closed unit ball B F ν(Ω) is a compact subset of F (Ω) and U a set of uniqueness for (T K , F ν).
Proof. We set X ∶= span{T K x x ∈ U }, Y ∶= F (Ω) and Z ∶= F ν(Ω). As in the proof of Theorem 3.7 we observe that X is σ( exists for each x ∈ U , implying the existence of lim ι S(A ι )(x) for each x ∈ X by linearity. We apply Proposition 7.1 and obtain f ∶= S(A) ∈ F ε ν(Ω, E) such that (A ι ) ι∈I converges to A in F (Ω)εE. From F (Ω) and F (Ω, E) being ε-into-compatible it follows that (f ι ) ι∈I converges to f in F (Ω, E).
First, we apply the preceding corollary γ-Hölder continuous functions. Similar to C 0,γ (Ω, E) we define the space of E-valued γ-Hölder continuous functions on Ω that vanish at a fixed point z ∈ Ω, but with a different topology. Let (Ω, d) be a metric space, z ∈ Ω, E an lcHs, 0 < γ ≤ 1 and define Then we have for every α ∈ A that z (Ω, E), z (Ω, E) with generator (T E , T K ). 7.3. Corollary. Let E be a Banach space, (Ω, d) a metric space with finite diameter, z ∈ Ω and 0 < γ ≤ 1. If (f ι ) ι∈I is a bounded net in C z (Ω) is a Banach space by [71,Proposition 1.6.2,p. 20]. For all f from the closed unit ball B F ν(Ω) of F ν(Ω) we have where diam(Ω) ∶= sup{d(x, y) x, y ∈ Ω} is the finite diameter of Ω. It follows that B F ν(Ω) is (uniformly) equicontinuous and {f (x) f ∈ B F ν(Ω) } is bounded in K for all x ∈ Ω. Ascoli's theorem (see e.g. [53,Theorem 47.1,p. 290]) implies the compactness of B F ν(Ω) in F (Ω). 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 F ε ν(Ω, E) = F ν(Ω, E) as linear spaces by Proposition 2.4 c). We note that lim ι f ι (x) = lim ι T E (f ι )(x, z) for all x in U , proving our claim by Corollary 7.2. [71] (see [ where Ω is compact, U = Ω and E = K. 7.4. Corollary. Let E be a Banach space, Ω ⊂ R d open and bounded, k ∈ N 0 and 0 < γ ≤ 1. In the case k ≥ 1, assume additionally that Ω has Lipschitz boundary. If (f ι ) ι∈I is a bounded net in C k,γ (Ω, E) such that exists for all 1 ≤ n ≤ d and x in a dense subset U ⊂ Ω, Ω is connected and there is x 0 ∈ Ω such that lim ι f ι (x 0 ) exists and k ≥ 1, then there is f ∈ C k,γ (Ω, E) such that (f ι ) ι∈I converges to f in C k (Ω, E).
then there is f ∈ C k,γ loc (Ω, E) such that (f ι ) ι∈I converges to f in C k (Ω, E). Proof. Let (Ω n ) n∈N be an exhaustion of Ω with open, relatively compact sets Ω n ⊂ Ω such that Ω n has Lipschitz boundary, Ω n ⊂ Ω n+1 for all n ∈ N and, in addition, x 0 ∈ Ω 1 and Ω n is connected for each n ∈ N in case (ii) (see the proof of Corollary 6.5). The restriction of (f ι ) ι∈I to Ω n is a bounded net in C k,γ (Ω n , E) for each n ∈ N. By Corollary 7.4 there is F n ∈ C k,γ (Ω n , E) for each n ∈ N such that the restriction of (f ι ) ι∈I to Ω n converges to F n in C k (Ω n , E) since U ∩ Ω n is dense in Ω n due to Ω n being open and x 0 being an element of the connected set Ω n in case (ii). The limits F n+1 and F n coincide on Ω n for each n ∈ N. Thus the definition f ∶= F n on Ω n for each n ∈ N gives a well-defined function f ∈ C k,γ loc (Ω, E), which is a limit of (f ι ) ι∈I in C k (Ω, E).
then there is f ∈ C k,1 loc (Ω, E) such that (f ι ) ι∈I converges to f in C k (Ω, E). Proof. By Corollary 6.5 b) (f ι ) ι∈I is a bounded net in C k,1 loc (Ω, E). Hence our statement is a consequence of Corollary 7.5.
The preceding result directly implies a C ∞ -smooth version. 7.7. Corollary. Let E be a Banach space and Ω ⊂ R d open. If (f ι ) ι∈I is a bounded net in C ∞ (Ω, E) such that (i) lim ι f ι (x) exists for all x in a dense subset U ⊂ Ω, or if (ii) lim ι (∂ en ) E f ι (x) exists for all 1 ≤ n ≤ d and x in a dense subset U ⊂ Ω, Ω is connected and there is x 0 ∈ Ω such that lim ι f ι (x 0 ) exists, then there is f ∈ C ∞ (Ω, E) such that (f ι ) ι∈I converges to f in C ∞ (Ω, E). Now, we turn to weighted kernels of hypoelliptic linear partial differential operators. 7.8. Corollary. Let E be a Banach space, Ω ⊂ R d open, P (∂) K a hypoelliptic linear partial differential operator, ν∶ Ω → (0, ∞) continuous and U ⊂ Ω a set of uniqueness for (Ω, E), τ co ). Proof. Our statement follows from Corollary 7.2 since by the proof of Corollary 3.10 all conditions needed are fulfilled.
A similar improvement of Corollary 7.3 for the space C z (Ω, E) of γ-Hölder continuous functions on a metric space (Ω, d) that vanish at a given point z ∈ Ω is possible, using the strict topology β on C z (Ω) given by the seminorms If Ω is compact and E a Banach space, this follows as in Corollary 7.9 from the observation that β is the mixed topology γ = γ( ⋅ C 0,γ (Ω) , τ co ) by [  Proof. Due to the proof of Corollary 3.12 all conditions needed to apply Corollary 7.2 are fulfilled, which proves our statement.

Wolff type results
The following theorem gives us a Wolff type description of the dual of F (Ω) and generalises [26, Theorem 3.3, p. 693] and its [26, Corollary 3.4, p. 694] whose proofs only need a bit of adaptation. 8.1. Theorem. Let F (Ω) and F (Ω, E) be ε-into-compatible, (T E , T K ) be a generator for (F ν, E) and a strong, consistent family for (F , E) for every Banach space E. Let F (Ω) be a nuclear Fréchet space and F ν(Ω) a Banach space whose closed unit ball B F ν(Ω) is a compact subset of F (Ω) and (x n ) n∈N fixes the topology in F ν(Ω). a) Then there is 0 < λ ∈ ℓ 1 such that for every bounded a n ν(x n )T K xn ∈ F ν(Ω) ′ a ∈ ℓ 1 , ∀ n ∈ N ∶ a n ≤ Cλ n }.
b) Let ( ⋅ k ) k∈N denote the system of seminorms generating the topology of F (Ω). Then there is a decreasing zero sequence (ε n ) n∈N such that for all Proof. We start with part a). Let for all f ∈ F ν(Ω) and a ∈ ℓ 1 it follows that E 1 is a linear subspace of F ν(Ω) ′ and the continuity of the map j 1 ∶ ℓ 1 → F ν(Ω) ′ where F ν(Ω) ′ is equipped with the operator norm. In addition, we deduce that the linear map j∶ ℓ 1 ker b ℓ 1 = [a] ℓ 1 ker j1 .
Let B be an absolutely convex, closed and bounded subset of F (Ω) ′ b . We endow W ∶= span B 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 ⊂ B W ′ , (µ k ) k∈N ⊂ V and 0 ≤ γ ∈ ℓ 1 such that which means that ρ ∈ ℓ 1 . For every µ ∈ B we set a n ∶= ∑ ∞ k=1 γ k w ′ k (µ)β (k) n , n ∈ N, and conclude that a ∈ ℓ 1 with a n ≤ ρ n for all n ∈ N and µ F ν(Ω) = ∞ n=1 a n ν(x n )T K xn .
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 sets (B l ) l∈N in F (Ω) ′ b by [52,Proposition 25.19,p. 303]. Due to our preceding results there is ρ (l) ∈ ℓ 1 with (8) for each l ∈ N. Finally, part a) follows from choosing 0 < λ ∈ ℓ 1 such that each ρ (l) is componentwise smaller than a multiple of λ, i.e. we choose λ in a way that there is C l ≥ 1 with ρ (l) n ≤ C l λ n for all n ∈ N. Let us turn to part b). We choose λ ∈ ℓ 1 from part a) and a decreasing zero sequence (ε n ) n∈N such that ( λn εn ) n∈N still belongs to ℓ 1 . For k ∈ N we set and note that the polarB ○ k is bounded in F (Ω) ′ b . Due to part a) there exists C ≥ 1 such thatÂ a n ν(x n )T K xn ∈ F ν(Ω) ′ a ∈ ℓ 1 , ∀ n ∈ N ∶ a n ≤ Cλ n }.

8.2.
Remark. The proof of Theorem 8.1 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 (F ν, 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(ℓ 1 ker j 1 ).
We recall from (5) that for a positive sequence ν ∶= (ν n ) n∈N and an lcHs E we have Further, we equip the space E N of all sequences in E with the topology of pointwise convergence, i.e. the topology generated by the seminorms for k ∈ N and α ∈ A.
Proof. We take F (N) ∶= K N and F (N, E) ∶= E N as well as F ν(N) ∶= ℓν(N) and since ℓν(N) = λ ∞ (A) with the Köthe matrix A ∶= (a n,k ) n,k∈N given by a n,k ∶= ν n for all n, k ∈ N. In addition, we have for every k ∈ N sup 1≤n≤k x n ≤ sup 1≤n≤k ν −1 n x ν ≤ sup 1≤n≤k ν −1 n , x = (x n ) n∈N ∈ B F ν(N) , which means that B F ν(N) is bounded in F (N). The space F (N) = K N is a nuclear Fréchet space and B F ν(N) is obviously closed in K N . Thus the bounded and closed set B F ν(N) is compact in F (N), implying our statement by Theorem 8.1.
b) Then there is a decreasing zero sequence (ε n ) n∈N such that for all compact K ⊂ Ω there is C ≥ 1 with Proof. Due to the proof of Corollary 3.10 and the observation that the space F (Ω) = (C ∞ P (∂) (Ω), τ co ) is a nuclear Fréchet space all conditions of Theorem 8.1 are fulfilled, which yields our statement.