Series representations in spaces of vector-valued functions

It is a classical result that every $\mathbb{C}$-valued holomorphic function has a local power series representation. This even remains true for holomorphic functions with values in a locally complete locally convex Hausdorff space $E$ over $\mathbb{C}$. Motivated by this example we try to answer the following question. Let $E$ be a locally convex Hausdorff space over a field $\mathbb{K}$, $\mathcal{FV}(\Omega)$ be a locally convex Hausdorff space of $\mathbb{K}$-valued functions on a set $\Omega$ and $\mathcal{FV}(\Omega,E)$ be an $E$-valued counterpart of $\mathcal{FV}(\Omega)$ (where the term $E$-valued counterpart needs clarification itself). For which spaces is it possible to lift series representations of elements of $\mathcal{FV}(\Omega)$ to elements of $\mathcal{FV}(\Omega,E)$? We derive sufficient conditions for the answer to be affirmative which are applicable for many classical spaces of functions $\mathcal{FV}(\Omega)$ having a Schauder basis. As a byproduct we obtain results on the representation of $\mathcal{FV}(\Omega,E)$ as a tensor product.


Introduction
The purpose of this paper is to lift series representations known from scalarvalued functions to vector-valued functions and its underlying idea was derived from the classical example of the (local) power series representation of a holomorphic function. Let D r ⊂ C be an open disc around zero with radius r > 0 and O(D r ) be the space of holomorphic functions on D r , i.e. the space of functions f ∶ D r → C such that the limit exists in C. It is well-known that every f ∈ O(D r ) can be written as where the power series on the right-hand side converges uniformly on every compact subset of D r and f (n) (0) is the n-th complex derivative of f at 0 which is defined from (1) by the recursion f (0) ∶= f and f (n) ∶= (f (n−1) ) (1) for n ∈ N. Amazingly by [8, 2.1 Theorem and Definition, p. [17][18] and [8,5.2 Theorem,p. 35], this series representation remains valid if f is a holomorphic function on D r with values in a locally complete locally convex Hausdorff space E over C where holomorphy means that the limit (1) exists in E and the higher complex derivatives are defined recursively as well. Analyzing this example, we observe that O(D r ), equipped with the topology of uniform convergence on compact subsets of D r , is a Fréchet space, in particular barrelled, with a Schauder basis formed by the monomials z ↦ z n . Further, the formulas for the complex derivatives of a C-valued resp. an E-valued function f on D r are built up in the same way by (1).
Our goal is to derive a mechanism which uses these observations and transfers known series representations for other spaces of scalar-valued functions to their vector-valued counterparts. Let us describe the general setting. We recall from [10, 14.2, p. 292] that a sequence (f n ) in a locally convex Hausdorff space F over a field K is called a topological basis, or simply a basis, if for every f ∈ F there is a unique sequence of coefficients (ζ K n (f )) in K such that where the series converges in F . Due to the uniqueness of the coefficients the map ζ K n ∶ f ↦ ζ K n (f ) is well-defined, linear and called the n-th coefficient functional corresponding to (f n ). Further, for each k ∈ N the map P K k ∶ F → F , P K k (f ) ∶= ∑ k n=1 ζ K n (f )f n , is a linear projection whose range is span(f 1 , . . . , f n ) and it is called the k-th expansion operator corresponding to (f n ). A basis (f n ) of F is called equicontinuous if the expansion operators P K k form an equicontinuous sequence in the linear space L(F, F ) of continuous linear maps from F to F (see [10, 14.3, p. 296]). A basis (f n ) of F is called a Schauder basis if the coefficient functionals are continuous which, in particular, is already fulfilled if F is a Fréchet space by [20,Corollary 28.11,p. 351]. If F is barrelled, then a Schauder basis of F is already equicontinuous and F has the bounded approximation property by the uniform boundedness principle.
The starting point for our approach is equation (2) . Let F and E be locally convex Hausdorff spaces over a field K where F is barrelled, has a Schauder basis (f n ) and F = F V(Ω) is a space of K-valued functions on a set Ω with a topology given by a family of weights V such that the point-evaluation functionals δ x , x ∈ Ω, are continuous. Let us suppose that there is a locally convex Hausdorff space F V(Ω, E) of functions from Ω to E such that the map is a topological isomorphism to its range where F V(Ω)εE ∶= L e (F V(Ω) ′ κ , E) is the ε-product of Schwartz. Assuming that for each n ∈ N and u ∈ F V(Ω)εE there is ζ E n (S(u)) ∈ E with ζ E n (S(u)) = u(ζ K n ), we can consider the (not necessarily convergent) series ∞ n=1 ζ E n (S(u))f n = ∞ n=1 u(ζ K n )f n in F V(Ω, E). In our main Theorem 3.8 we state sufficient conditions such that this series converges to S(u) in F V(Ω, E) for every u ∈ F V(Ω)εE, i.e. we lift the series representation (2) from the scalar-valued case to the range of S. If S is even surjective, we reach our goal of lifting to the vector-valued case. For example in the case of holomorphic functions, S is surjective as a map from O(D r )εE to O(D r , E) for locally complete E by [2, Theorem 9, p. 232]. We apply our result to sequence spaces, spaces of continuously differentiable functions on a compact interval, the Schwartz space and the space of smooth functions which are 2π-periodic in each variable.
Our main theorem has two byproducts. One of them is the answer to the question when F V(Ω)εE resp. F V(Ω, E) has the bounded approximation property. The second byproduct is that the completion of the injective tensor product F V(Ω)⊗ ε E is isomorphic to F V(Ω)εE under suitable mild assumptions, especially, that every element of F V(Ω)⊗ ε E has a series representation as well. Concerning series representation in F V(Ω)⊗ ε E, little seems to be known whereas for the completion F⊗ π E of the projective tensor product F ⊗ π E of two metrisable locally convex spaces F and E it is well-known that every f ∈ F⊗ π E has a series representation f = ∞ n=1 λ n f n ⊗ e n where (λ n ) ∈ ℓ 1 , i.e. (λ n ) is absolutely summable, and (f n ) and (e n ) are null sequences in F and E, respectively (see e.g. [9, Chap. I, §2 , n ○ 1, Théorème 1, p. 51] or [10,15.6.4 Corollary,p. 334]). If F and E are metrisable and one of them is nuclear, then the isomorphy F⊗ π E ≅ F⊗ ε E holds and we trivially have a series representation of the elements of F⊗ ε E as well. Other conditions on the existence of series representations of the elements of F⊗ ε E can be found in [22,Proposition 4.25,p. 88], where F and E are Banach spaces and both of them have a Schauder basis, and in [11,Theorem 2,p. 283], where F and E are locally convex Hausdorff and both of them have an equicontinuous Schauder basis.

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 topological vector space X, we write acx(M ) for the closure of the absolutely convex hull acx(M ) of M in X. By E we always denote a non-trivial locally convex Hausdorff space, in short lcHs, over the field K = R or C equipped with a directed fundamental system of seminorms (p α ) α∈A . If E = K, then we set (p α ) α∈A ∶= { ⋅ }. We recall that 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 gauge functional of D as a norm (see [10, p. 151]). The space E is called locally complete if E D is a Banach space for every closed disk D ⊂ E (see [10,10.2.1 Proposition,p. 197]). For details on the theory of locally convex spaces see [7], [10] or [20]. By X Ω we denote the set of maps from a non-empty set Ω to a non-empty set X, by χ K the characteristic function of a subset K ⊂ Ω 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. If F and E are (linearly topologically) isomorphic, we write F ≅ E and, if F is only isomorphic to a subspace of E, we write F↪E. We denote by L t (F, E) the space L(F, E) equipped with the locally convex topology 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 precompact (totally bounded) subsets of F if t = γ and on the bounded subsets of F if t = b. 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 [25, 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 (3) 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 . Jarchow uses a third, different definition of the ε-product (see [10, 16.1, p. 344]) which coincides with the one of Schwartz if F is complete by [10,9.3.7 Proposition,p. 179]. However, we stick to Schwartz' definition. For locally convex Hausdorff spaces F i , E i and T i ∈ L(F i , E i ), i = 1, 2, we define the ε-product T 1 εT 2 ∈ L(F 1 εF 2 , E 1 εE 2 ) of the operators T 1 and T 2 by is the dual map of T 1 . If T 1 and T 2 are isomorphies, then T 1 εT 2 is an isomorphy into, i.e. T 1 εT 2 is an isomorphy from F 1 εE 1 to its range by [ In particular, if E has BAP, then it has AP (see [12,Approximationseigenschaft,p. 247]). If E and F are locally convex Hausdorff spaces such that E has BAP and E ≅ F via an isomorphism Ψ, then F also has BAP since it is easily checked that the net formed by For more information on the theory of ε-products and tensor products see [6], [10] and [12].

The tensorproduct for weighted function spaces
First, we introduce the spaces F V(Ω, E) mentioned in our first section and recall some basic definitions and results from [15,Section 3]. To the end of this section we state our main theorem which gives a sufficient condition for series representations of the elements of F V(Ω)εE resp. F V(Ω, E) derived from series representations in F V(Ω). We start with the definition of a family of weights.
3.1. Definition (weight function [15,3.1 Definition,p. 3]). Let Ω, J, L be nonempty sets and (M l ) l∈L a family of non-empty sets. V ∶= ((ν j,l,m ) m∈M l ) j∈J,l∈L is called 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 ker T E m as well as Further, we write F V(Ω) ∶= F V(Ω, K). If we want to emphasise dependencies, we write M(F V) and M(E) instead of M and the same for M top , M 0 and M r .
The space F V(Ω, E) is a locally convex Hausdorff space by (4) and we call it a dom-space if its system of seminorms is directed and, additionally, is well-defined. If we want to emphasize the dependence of S on F V(Ω), we write S F V(Ω) instead of S. The next definition describes a sufficient condition for the To be precise, T K m,x in (i) and (ii) means the restriction of T K m,x to F V(Ω). The consistency of a family of operators yields to the following theorem.
3.4. Theorem ([15, 3.7 Theorem, p. 6]). Let (T E m , T K m ) m∈M be a consistent family for (F V, E). Then S∶ F V(Ω)εE → F V(Ω, E) is an isomorphism into. Now, we phrase a sufficient condition such that F V(Ω)εE ≅ F V(Ω, E) holds. As usual we consider F ⊗ E as an algebraic subspace of F εE for two locally convex Hausdorff spaces F and E by means of the linear injection We omit the index F and just write χ if no confusion seems to be likely. Via χ the topology of F εE induces a locally convex topology on F ⊗ E and F ⊗ ε E denotes F ⊗ E equipped with this topology.
3.6. Remark. If the conditions of Corollary 3.5 b) in the 'complete-dense'-case are satisfied for every Banach space E or for E = F V(Ω) ′ κ , then F V(Ω) has AP by [12,Satz 10.17,p. 250].
Let us prepare the ground for our main theorem.
where the series converges in F V(Ω, E). If F V(Ω) and E are complete (resp. quasi-, sequentially complete), a) then with * = c (resp. qc, sc) where the series converges in F V(Ω)⊗ * ε E. b) the topology on F V(Ω, E) is stronger than the topology of pointwise convergence and then the inverse of S is given by the map Proof. Let f ∈ F V(Ω, E) and observe that for every k ∈ N by Corollary 3.5 a). Due to our assumption we have P E k (f ) → f in F V(Ω, E). Thus F V(Ω) ⊗ E is sequentially dense in F V(Ω, E). The remaining part of a), in particular that S is an isomorphism, follows from Corollary 3.5 b). Let us turn to part b). Let f ∈ F V(Ω, E), j ∈ J and l ∈ L. Then for every e ′ ∈ E ′ , x ∈ Ω, m ∈ M l and n ∈ N. Hence for e ′ ∈ E ′ there are α ∈ A and C > 0 such that k n=q e ′ (e n (f ))f n j,l = e n (f )f n j,l,α for all k, q ∈ N, k > q, which implies that (∑ k n=1 e ′ (e n (f ))f n ) is a Cauchy sequence in F V(Ω) and thus convergent. Furthermore, we notice that for every x ∈ Ω where we used in the first equality that the topology on F V(Ω, E) is stronger than the topology of pointwise convergence and the same for F V(Ω) in the second equality which is a consequence of F V(Ω) being a dom-space. Therefore we deduce that for every y ∈ F V(Ω) ′ . We denote bŷ for every e ′ ∈ E ′ . We derive that Let us turn to our main theorem which can be combined with the preceding proposition.
3.8. Theorem. Let F V(Ω) be barrelled and (T E m , T K m ) m∈M a consistent family for (F V, E) which fulfils: Let (T E mn , T K mn ) n∈N be a subfamily, x n ∈ ω mn for every n ∈ N and (f n ) n∈N be a sequence in F V(Ω) such that where the series converges in F V(Ω). Then the following holds. a) F V(Ω) has BAP, F V(Ω) ⊗ E is sequentially dense in F V(Ω)εE and where the series converges in F V(Ω, E). b) If F V(Ω) and E are complete (resp. quasi-, sequentially complete), then where the series converges in F V(Ω)⊗ * ε E with * = c (resp. qc, sc). c) If S is surjective, then (5) holds with e n (f ) = T E mn,xn (f ). d) F V(Ω)εE has BAP if F V(Ω)εE is barrelled and the restriction of T E mn,xn to F V(Ω, E) is continuous for every n ∈ N.

Proof.
a) For k ∈ N we define the linear projection The range of P K k is finite dimensional (dim ≤ k). Due to the consistency of the family (T E m , T K m ) m∈M we have T K mn,xn ∈ F V(Ω) ′ and thus for every j ∈ J and l ∈ L there are C k > 0, j k ∈ J and l k ∈ L such that is equicontinuous by the uniform boundedness principle. Therefore F V(Ω) has BAP. By virtue of the Banach-Steinhaus theorem we get P K k → id in L κ (F V(Ω), F V(Ω)). Moreover, we deduce from T K mn,xn ∈ F V(Ω) ′ for every n ∈ N that P K k,x ∈ F V(Ω) ′ . Let x ∈ Ω. Since F V(Ω) is a dom-space, there are j ∈ J, l ∈ L and C > 0 such that for all k ∈ N Hence the convergence of (P K k (f )) to f in F V(Ω) implies that (P K k,x ) converges to δ x in F V(Ω) ′ σ . We conclude that (P K k,x ) converges to δ x in F V(Ω) ′ κ from the Banach-Steinhaus theorem as F V(Ω) is barrelled. We define the linear map This map is well-defined and for every e ∈ E and f n ∈ F V(Ω) by Corollary 3.5 a). Using the consistency, we obtain by a simple calculation for every u ∈ F V(Ω)εE and x ∈ Ω that Hence for every α ∈ A there are C = C(u) > 0 and compact for every j ∈ J and l ∈ L.
Follows from a) and Corollary 3.5 a). c) Follows directly from a). d) The map P E k from a) has finite rank and its range is contained in F V(Ω)εE. Furthermore, we have for every f ∈ F V(Ω, E), m ∈ M top and x ∈ Ω yielding to T E mn,xn (f )f n j,l,α = f n j,l p α T E mn,xn (f ) for every j ∈ J, l ∈ L, α ∈ A and n ∈ N. Due to the continuity of the restriction of every T E mn,xn to F V(Ω, E) we obtain that P E k is continuous. (10). Like before it follows that the restriction of the sequence (P E k ) k∈N to F V(Ω)εE is equicontinuous by the uniform boundedness principle and F V(Ω)εE has BAP since F V(Ω)εE is barrelled.
3.9. Remark. The 'continuity-condition' in part d) is fulfilled if m n ∈ M top for every n ∈ N. Indeed, then there are l ∈ L with m n ∈ M l and j ∈ J such that ν j,l,mn (x n ) > 0 by (4) and it follows for every f ∈ F V(Ω, E) and α ∈ A connoting the continuity of T E mn,xn .
The first part of the proof of Theorem 3.8 a) is just an adaption of the proof of [10, 18.5.1 Proposition, p. 410] where it is shown that a locally convex Hausdorff space which admits an equicontinuous basis has Grothendieck's approximation property. If (f n ) is a topological basis of F V(Ω), then it is a Schauder basis due to the continuity of the coefficients T K mn,xn . A stability result for AP like Theorem 3.8 d) for BAP is known for quasi-complete locally convex Hausdorff spaces F and E if both of them have AP by [14, §44, 5.(7) b), p. 284]. Concerning the condition of F V(Ω)εE being barrelled in part d), we recall the definition of an L ∞ -space and of fundamental- Proof. The first statement follows from the fact that the ε-product of two Fréchet spaces is also Fréchet by [ As mentioned in the introduction, every element of has a series representation if F V(Ω) and E are metrisable and one of the spaces is nuclear. In the subsequent considerations we try to relax the conditions on F V(Ω) and E a little bit. We recall that the projective topology on the tensor product F V(Ω)⊗E is stronger than the injective topology by [10, 16.1 a) If there are a metrisable locally convex space G, a continuous, linear, surjective map P E ∶ G⊗ π E → F V(Ω, E) and a continuous, linear map where I G ∶ G ⊗ π E → G⊗ π E is the embedding in the completion, then Since P E is surjective and G and E are metrisable locally convex spaces, there is g ∈ G⊗ π E such that f = P E (g) and where (λ n ) ∈ ℓ 1 and (g n ) and (e n ) are null sequences in G resp. E by virtue of [10, 15.6.4 Corollary, p. 334] (the map I G is usually omitted). Denoting by the embedding in the completion and using the linearity and continuity of P E , we derive where we used [10, 3.4.2 Theorem, p. 61-62] and that F V(Ω)εE is complete for the last equation. We claim that (∑ k n=1 λ n P K (g n ) ⊗ e n ) is a Cauchy sequence in F V(Ω) ⊗ π E. Indeed, since P K is linear and continuous, it follows that (P K (g n )) is a null sequence in F V(Ω) as well. Hence for j ∈ J, l ∈ L and α ∈ A there is C > 0 such that to an element h f and we obtain by the continuity of S ○χ π that Since F V(Ω)εE is complete, the completion of F V(Ω) ⊗ ε E is obtained as the closure in F V(Ω)εE. We observe thatχ π (h f ) ∈ F V(Ω)εE and obtain from the continuity ofχ π and the equationχ π ○ I F = id ○χ that Therefore S is surjective and we get by Theorem 3.4 Furthermore, for every f ∈ F V(Ω)⊗ c ε E we have by (12) and (13) that with some g ∈ (P E ) −1 (S(f )) from (11). Let us turn to part b). Due to (12) is continuous, linear and surjective. It is also injective because S andχ π are injective. From the open mapping theorem (De Wilde) we conclude that the inverse of S ○χ π is continuous as is a Banach space with AP and E a Banach space due to [6, 5.6 Theorem, p. 64] or more general • every local Banach space F V(Ω) ⋅ j,l , j ∈ J, l ∈ L, (see [20,Definition,p. 279]) has AP by [

Applications
Before we apply our results from the preceding section, we remark the following observation which comes in handy quite often.
to the defining family for (F V, E), we do not change the spaces F V(Ω, Y ). Further, the new, bigger defining family is still consistent since we have for every u ∈ F V(Ω)εE that In some of the references on Schauder basis used in this section the given Schauder bases are bases for real spaces of functions. They can be turned into Schauder bases for corresponding complex spaces of functions by the following procedure (which we implicitely do if needed).
we can consider the f n as functions in F V(Ω, C) via the map f n ↦ f n +i⋅0 and they form a Schauder basis of F V(Ω, C) with associated coefficient , which is easily seen.

Sequence spaces
For our first application we recall the definition of some sequence spaces. A matrix A ∶= (a k,j ) k,j∈N of non-negative numbers is called Köthe matrix if it fulfils: For an lcHs E we define the Köthe space and the topological subspace In particular, the space c 0 (N, E) of null-sequences in E is obtained as converges in E} and equipped with the system of seminorms for α ∈ A. We define the spaces of E-valued rapidely decreasing sequences which we need for our subsection on Fourier expansion by Furthermore, we equip the space E N with the system of seminorms given by for l ∈ N and α ∈ A. For a non-empty set Ω we define for n ∈ Ω the n-th unit function by and we simply write ϕ n instead of ϕ n,Ω if no confusion seems to be likely. Further, we set ϕ ∞ ∶ N → K, ϕ ∞ (k) ∶= 1, and x ∞ ∶= lim k→∞ x k for x ∈ c(N, E). For series representations of the elements in these sequence spaces we do not need our main Theorem 3.8 due to the subsequent proposition but we obtain some additional structural properties of the sequence spaces from it.

4.3.
Proposition. Let E be an lcHs and ℓV(Ω, E) one of the spaces where the series converges in ℓV(Ω, E). b) Then where the series converges in c(N, E).
Proof. Let us begin with a). For x = (x n ) ∈ ℓV(Ω, E) let (s m ) be the sequence in ℓV(Ω, E) given by s m ∶= ∑ n ≤m x n ϕ n . Let ε > 0, α ∈ A and j ∈ N. For x ∈ c 0 (A, E) there is N 0 ∈ N such that p α (x n a n,j ) < ε for all n ≥ N 0 . Hence we have for Thus we deduce for n ≥ N 1 and hence x − s m j,α = sup n >m p α (x n )a n,j ≤ sup n ≥N1 p α (x n )(1 + n 2 ) j 2 ≤ ε x 2j,α for all m ≥ N 1 . Therefore (s m ) converges to x in ℓV(Ω, E) and Theorem. Let E be an lcHs and ℓV(Ω, E) one of the spaces c 0 (A, E), E N , for all x ∈ ℓV(Ω, E) and which implies T K lim,1 ∈ c(N) ′ and δ n → T K lim,1 in c(N) ′ γ by the Banach-Steinhaus theorem since c(N) is a Banach space. Hence we get for every u ∈ c(N)εE. Therefore (T E lim , T K lim ) is a consistent subfamily for (c, E). The whole defining family of (c, E), which we get by adding (T E id , T K id ) to the subfamily, is clearly consistent. From Proposition 4.3 and Proposition 3.7 a) we deduce part a). Let us turn to part b). The spaces ℓV(Ω) and c(N) are Fréchet spaces and thus barrelled. In addition, the subfamily (T E id , T K id ) of the defining families fulfils (8).
x n for n ∈ Ω and Y ∈ {K, E}, we remark that we do not change the spaces ℓV(Ω, Y ) if we add Ω to M r and the family (T E n , T K n ) n∈Ω to the defining families of these spaces. The new, bigger defining families are still consistent by Remark 4.1 applied to (T E id , T K id ) and z = n. Defining E}, we remark that we do not change the spaces c(N, Y ) if we add N to M r and the family (T E n ,T K n ) n∈N to the defining family of this space. The new, bigger defining family is still consistent sincẽ ) for every u ∈ c(N)εE and n ∈ N. We deduce from Proposition 4.3 that for all x ∈ ℓV(Ω) and for all x ∈ c(N). If follows from Remark 4.1 that the restrictions of T E n,1 to ℓV(Ω, E) are continuous for every n ∈ Ω. For c(N, E) we observe that (N, E), for every α ∈ A and n ∈ N. Therefore part b) follows from a), Theorem 3.8 d) and

Continuous and differentiable functions on a closed interval
We start with continuous functions on compact sets. We recall the following definition from [27, p. 259]. A locally convex Hausdorff space is said to have the metric convex compactness property (metric ccp) if the closure of the absolutely convex hull of every metrisable compact set is compact. In particular, every sequentially complete space has metric ccp and this implication is strict (see [15, p. 17-18] and the references therein). Let E be an lcHs, Ω ⊂ R d be compact and denote by C(Ω, E) ∶= C 0 (Ω, E) the space of continuous functions from Ω to E. We equip C(Ω, E) with the system of seminorms given by 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 if the right-hand side exists in E for every x ∈ Ω.
are k-times continuously partially differentiable on Ω and whose partial derivatives up to order k are continuously extendable on Ω. We equip C k (Ω, E) with the system of seminorms given by for α ∈ A. Let us introduce the operators that describe the continuous extendability. First, we set , β ≤k and it holds C k (Ω, E) ≅ C k (Ω)εE via S. We want to apply our preceding results to intervals. Let −∞ < a < b < ∞ and T ∶= (t j ) 0≤j≤n be a partition of the interval [a, b], i.e. a = t 0 < t 1 < . . . < t n = b. The hat functions h T tj ∶ [a, b] → R for the partition T are given by Let T ∶= (t n ) n∈N0 be a dense sequence in [a, b] with t 0 = a, t 1 = b and t n ≠ t m for n ≠ m. For T n ∶= {t 0 , . . . , t n } there is a (unique) enumeration σ∶ {0, . . . , n} → T n such that T n ∶= (t σ(j) ) 0≤j≤n is a partition of [a, b]. The functions ϕ T 0 ∶= h T1 t0 , ϕ T 1 ∶= h T1 t1 and ϕ T n ∶= h Tn tn for n ≥ 2 are called Schauder hat functions for the sequence T and form a Schauder basis of C([a, b]) with associated coefficient functionals given by where we identified f (k) with its continuous extension on [a, b]. The resulting Schauder basis f T n ∶ [a, b] → R and associated coefficient functionals  ([a, b], E) and so we define µ E n on C k ([a, b], E) for n ∈ N 0 accordingly. 4.9. Theorem. Let E be an lcHs, k ∈ N, T ∶= (t n ) n∈N0 be a dense sequence in [a, b] with t 0 = a, t 1 = b and t n ≠ t m for n ≠ m. a) If E has metric ccp, then where the series converges in C k ([a, b], E). b) If E is complete (resp. quasi-, sequentially complete), then Proof. The Banach space C k ([a, b]) is barrelled. The defining family for (C k , E) is consistent by Proposition 4.7 and the subfamily ((∂ n ) E , (∂ n ) K ) 0≤n≤k clearly satisfies (8). We do not change the spaces C k ([a, b], Y ), Y ∈ {K, E}, if we add N 0 to M r and (µ E n , µ K n ) n∈N0 to the defining family for (C k ([a, b]), E). The new defining family is still constistent since the subfamily (µ E n , µ K n ) n∈N0 only consists of (finite) linear combinations of point evaluations of derivatives of order k at inner points of [a, b] or of limits to the end points a or b of point evaluations of derivatives up to order k. To see that the latter are consistent we set , with t ∈ {a, b} for 0 ≤ n ≤ k and Y ∈ {K, E} and use Remark 4.1 applied to ((∂ n ) E ext , (∂ n ) K ext ) and z = t. The maps T E (t,n),1 are also continuous because for α ∈ A and 0 ≤ n ≤ k. In combination with Remark 4.1 applied to ((∂ k ) E , (∂ k ) K ) and z = t m ∈ (a, b), m ≥ 2, we derive the consistency of (µ E n , µ K n ) n∈N0 and the continuity of every µ E n . Since S C k ([a,b]) is surjective by Proposition 4.7 if E has metric ccp, we deduce part a) and b) from Theorem 3.8 c) and Proposition 3.7 a). Part c) follows from the continuity of the operators µ E n , Theorem 3.8 d) and Remark 3.10 a) because C k ([a, b]) is a Banach space.

Fourier expansions
In this subsection we turn our attention to Fourier expansions in the space of vector-valued rapidely decreasing functions and in the space of vector-valued smooth, 2π-periodic functions. We start with the definition of the Pettis-integral which we need to define the Fourier coefficients for vector-valued functions. For a measure space (X, Σ, µ) and 1 ≤ p < ∞ let µ). From now on we do not distinguish between equivalence classes and their representants anymore. For a measure space (X, Σ, µ) and f ∶ X → K we say that f is integrable on Λ ∈ Σ and write f ∈ L 1 (Λ, µ) if χ Λ f ∈ L 1 (X, µ). Then we set

4.10.
Definition (Pettis-integral). Let (X, Σ, µ) be a measure space and E an lcHs. A function f ∶ X → E is called weakly (scalarly) measurable if the function In this case e Λ is unique due to E being Hausdorff and we set If we consider the measure space (X, L (X), λ) of Lebesgue measurable sets for X ⊂ R d , we just write dx ∶= dλ(x).

4.11.
Lemma. Let E be a locally complete lcHs, Ω ⊂ R d open and f ∶ Ω → E. If f is weakly C 1 , i.e. e ′ ○ f ∈ C 1 (Ω) for every e ′ ∈ E ′ , then f is Pettis-integrable on every compact subset of K ⊂ Ω with respect to any locally finite measure µ on Ω and Proof. Let K ⊂ Ω be compact and (Ω, Σ, µ) a measure space with locally finite measure µ, i.e. Σ contains the Borel σ-algebra B(Ω) on Ω and for every x ∈ Ω there is a neighbourhood U x ⊂ Ω of x such that µ(U x ) < ∞. Since the map e ′ ○ f is differentiable for every e ′ ∈ E ′ , thus Borel-measurable, and B(Ω) ⊂ Σ, it is measurable. We deduce that e ′ ○ f ∈ L 1 (K, µ) for every e ′ ∈ E ′ because locally finite measures are finite on compact sets. Hence the map is well-defined and linear. We estimate Due to f being weakly C 1 and [3, Proposition 2, p. 354] the absolutely convex set acx(f (K)) is compact yielding I ∈ (E ′ κ ) ′ ≅ E by the theorem of Mackey-Arens which means that there is e K ∈ E such that where we used [20,Proposition 22.14,p. 256] in the first and last equation to get from p α to sup e ′ ∈B ○ α and back. For an lcHs E we define the Schwartz space of E-valued rapidely decreasing functions by We recall the definition of the Hermite functions. For n ∈ N 0 we set with the Hermite polynomials H n of degree n which can be computed recursively by

4.12.
Proposition. Let E be a sequentially complete lcHs, f ∈ S(R d , E) and n ∈ N d 0 . Then f h n is Pettis-integrable on R d .
Proof. For k ∈ N we define the Pettis-integral which is a well-defined element of E by Lemma 4.11. We claim that (e k ) is a Cauchy sequence in E. First, we notice that there are j ∈ N and C > 0 such that H n (x) ≤ C(1 + x 2 ) j 2 for all x ∈ R d as H n is a product of polynomials in one variable. Now, let α ∈ A, k, m ∈ N, k > m, and set proving our claim. Since E is sequentially complete, the limit y ∶= lim k→∞ e k exists in E. From e ′ ○ f ∈ S(R d ) for every e ′ ∈ E ′ and the dominated convergence theorem we deduce which yields the Pettis-integrability of f h n on R d with ∫ R d f (x)h n (x)dx = y.
Due to the previous proposition we can define the n-th Fourier coefficient of if E is sequentially complete. We know that the map is an isomorphism and its inverse is given by is an isomorphism. b) If E is complete (resp. quasi-, sequentially complete), then for * = c (resp. qc, sc) and for every n ∈ N d 0 and e ′ ∈ E ′ . Thus we have for every e ′ ∈ E ′ which implies by [20, Mackey's theorem 23.15, p. 268 is consistent by [15, 5.10 Example a), p. 24]. We notice that by Theorem 4.4 a)(i), the second isomorphism into (i.e. to its range) is the map (F K ) −1 ε id E and the third isomorphism into is the map S S(R d ) by Theorem 3.4. Next, we show that is surjective and the inverse of F E . We can explicitely compute the composition of these maps. By Proposition 3.7 b) we get that the inverse of S s(N d 0 ) is given by which gives yielding the surjectivity of the composition. Therefore is an isomorphism with right inverse F E which implies that F E is its inverse proving part a). In addition, the bijectivity of for every f ∈ S(R d , E), y ∈ S(R d ) ′ and e ′ ∈ E ′ which implies to the defining family for (S, E), we do not change the spaces. Next, we show that the new, bigger family is still consistent. Let n ∈ N d 0 and u ∈ S(R d )εE. Then f ∶= S(u) ∈ S(R d , E) and thus S(u) ∈ dom F E n . Let j ∈ N. Thanks to the continuity of F K there are C > 0 and l ∈ N such that which results in As a consequence the map (F K ) −1 ε id E is also an ismorphism (onto) if E is sequentially complete. The first isomorphy in part b) generalises [15, 5.11 Example, p. 25] from quasi-complete to sequentially complete spaces. Our last example of this subsection is devoted to Fourier expansions of vector-valued 2π-periodic smooth functions. We equip the space C ∞ (R d , E) for locally convex Hausdorff E with the system of seminorms generated by for K ⊂ R d compact, l ∈ N 0 and α ∈ A. By C ∞ 2π (R d , E) we denote the topological subspace of C ∞ (R d , E) consisting of the functions which are 2π-periodic in each variable. Being 2π-periodic can be described by defining operators as well. For a function g∶ R d → E being 2π-periodic in each variable is equivalent to g ∈ ker T E Lemma 4.11 we are able to define the n-th where ⟨⋅, ⋅⟩ is the usual scalar product on R d , if E is locally complete.
4.14. Theorem. Let E be an lcHs over C.
where the series converges in C ∞ 2π (R d , E). b) If E is complete (resp. quasi-, sequentially complete), then where the series converges in The whole defining family which we get by adding γ ≤k} to the subfamily is consistent by [15, 5.10 Example a), p. 24]. Since (T E per , T C per ) is also a strong subfamily in the sense of [15,3.9 Definition,p. 6] [15, 5.12 Example, p. 25] and [15, 3.19 Proposition (i), p. 11]. By [15, 3.14 Theorem, p. 8] the inverse of S is given by 2π , E), we do not change the spaces. Next, we show that the new, bigger family is still consistent. Let n ∈ Z d and u ∈ C ∞ 2π (R d )εE. Then f ∶= S(u) ∈ C ∞ 2π (R d , E) and thus S(u) ∈ dom F E n . Further, the estimate F C n,1 (g) = ĝ(n) ≤ sup which results in (1) and thus proves the consistency. Every f ∈ C ∞ 2π (R d ) can be written as where the series converges in the topology of C ∞ 2π (R d ) (see e.g. [12,Satz 1.7,p. 18]). Moreover, C ∞ 2π (R d ) is a Fréchet space, thus barrelled, and it is easily verified that subfamily ((∂ β ) E , (∂ β ) C ) β∈N d 0 of partial derivatives of the (new) defining family for (C ∞ 2π , E) fulfils (8). Hence we get from Theorem 3.8 c) that we can apply Proposition 3.7 a) proving a) and b). Part c) follows from Theorem 3.8 d) since we have for every α ∈ A by Lemma 4.11 Considering the coefficients in the series expansion above, we know that the map is an isomorphism (see e.g. [12,Satz 1.7,p. 18]). Thus we have the following relation if E is a locally complete Hausdorff space over C , the second isomorphism into is the map F C ε id E and the third isomorphism into is the map S s(Z d ) by Theorem 3.4. We can explicitely compute the composition of these maps. With the notation from the proof above we have for every

Thus the map
well-defined and an isomorphism into if E is locally complete. If E is sequentially complete, it is even an isomorphism to the whole space s(Z d , E).

4.15.
Theorem. If E is a sequentially complete lcHs over C, Proof. Due to our previous considerations we only need to show that F E is surjective. Let a ∶= (a k ) k∈Z d ∈ s(Z d , E) and set G n (a) ∶= ∑ k ≤n a k e i⟨k,⋅⟩ for n ∈ N 0 . Then (G n (a)) is a sequence in C ∞ 2π (R d , E). Let K ⊂ R d be compact, l ∈ N 0 and α ∈ A. For every n, m ∈ N 0 , m < n, we have εE and E is sequentially complete. Therefore (G n (a)) converges in C ∞ 2π (R d , E) and we set G(a) ∶= ∑ k∈Z d a k e i⟨k,⋅⟩ . For every e ′ ∈ E ′ we get e ′ ○ G(a) = ∑ k∈Z d e ′ (a k )e i⟨k,⋅⟩ which yields e ′ ○ G(a)(n) = e ′ (a n ) for every n ∈ Z d by [12,Satz 1.7,p. 18]. Hence we obtain for every e ′ ∈ E ′ and n ∈ Z d and thus G(a)(n) = a n implying F E (G(a)) = a.
We remark that this result implies that F C ε id E is also an isomorphism (onto) if E is sequentially complete since S s(Z d ) is an isomorphism by Theorem 4.4 a)(i). This is used without a proof in [12,Satz 10.8,p. 239] to obtain the corresponding result for quasi-complete E which we generalised here. It is an open problem whether it is still true if E is only locally complete. If one could show that (G n (a)) is a local Cauchy sequence, then its convergence would follow from the fact that C ∞ 2π (R d , E) is locally complete if E is locally complete.
is a nuclear Fréchet space as a closed subspace of the nuclear Fréchet space is an isomorphism implying that s(Z d ) is nuclear as well. We observe that for every f ∈ C ∞ 2π (R d ) and e ∈ E by Corollary 3.5 a) which yields to

Duality method
In our last subsection we want to apply our method to spaces which are not spaces in the sense of Section 3 but which are isomorphic to strong duals of nice locally convex spaces (lcs). The underlying idea was derived from considering separable Hilbert spaces and the Riesz representation theorem. Let F be an lcHs over K with a Schauder basis (f n ) and coefficient functionals (ζ K n ). Suppose there is an isomorphism Ψ∶ F → X ′ b where X is a normed or semi-reflexive, metrisable locally convex space over K. Then X ′ b = L b (X, K) is barrelled and complete as the strong dual of a normed space is a Banach space and the strong dual of a semi-reflexive, metrisable locally convex space is a barrelled, complete (DF)-space by [10, 11.4 . A function g∶ X → E is linear if and only if g ∈ ker T E a ∩ ker T E h (additive and homogeneous). The defining family (T E m , T K m ) m∈{c}∪{a,h} for (L b , E) is consistent by [15,4.3 Proposition,p. 13] for Ω = X in combination with [15, 4.5 Remark (ii), p. 15] for continuity and by [15, 4.14 Proposition, p. 17] for linearity. In addition, the subfamily (T E c , T K c ) of the defining family fulfils (8). A Schauder basis of X ′ b is now given by (Ψ(f n )) and the associated coefficient , for every n ∈ N. Now, in order to apply Theorem 3.8 we only have to guarantee that for each n ∈ N there is a linear operator T E n ∶ L b (X, E) → E {1} such that we do not change the spaces L b (X, Y ), Y ∈ {K, E}, if we add N to M r and (T E n , T K n ) n∈N to the defining family for (L b , E) and that the new, bigger defining family is still consistent. This can be assured by assuming that (f n ) is also a sequence in X and that the coefficient functionals under this assumption and by setting T E n (y)(1) ∶= y(f n ), y ∈ L b (X, E), for each n the consistency of the new defining family follows from Remark 4.1 applied to (T E c , T K c ) and z = f n . The same remark yields the continuity of every T E n as well and hence we have just arrived at the following.
4.17. Theorem. Let E be an lcHs, let F be an lcHs with a Schauder basis (f n ) and associated coefficient functionals (ζ K n ), let X be a normed or semi-reflexive, metrisable lcs such that (f n ) ⊂ X and there is an isomorphism Then the following holds.
a) X ′ b ⊗ E resp. F ⊗ E is sequentially dense in X ′ b εE resp. F εE and where the series converge in L b (X, E) resp. F εE. b) If E is complete (resp. quasi-, sequentially complete), then where the series converge in X ′ b⊗ * ε E resp. F⊗ * ε E. c) If E is a semi-Montel (resp. complete semi-Montel) space, then where the series converges in L b (X, E). d) If X is normed and E a Fréchet space, then X ′ b εE and F εE have BAP. e) If F is an L ∞ -space and E an (LF)-space or if E is locally complete, barrelled and the strong dual E ′ b is fundamentally-ℓ 1 -bounded, then X ′ b εE and F εE have BAP.
Proof. Due to our preceding considerations and Theorem 3.8 we obtain that a)+b) and d)+e) hold for X ′ b . In part c) we notice that semi-Montel spaces are already quasi-complete. Then c) is a consequence of part b) since S X ′ b is surjective if E is a semi-Montel space by [15,5.6 Example,p. 20]. Let us turn to part a) for F . We set Q k ∶ F εE → F εE, Q k (u) ∶= ∑ k n=1 χ F f n ⊗u(ζ K n ) , and observe that the range of Q k is contained in F ⊗ E and Q k has finite rank for every k ∈ N. From for every x ∈ X and for every u ∈ F εE. Since S X ′ b ○Ψε id E is an isomorphism into, this implies Q k (u) → u in F εE and thus part a) for F . Part b) for F is a direct consequence of part a) for F . Concerning part d) and e) for F , we observe that F is a Banach space in both cases whose norm we denote by ⋅ . Let α ∈ A and K ⊂ F ′ be equicontinuous. Then there is C > 0 such that f ′ (f n ) ≤ C f n for all n ∈ N and f ′ ∈ K. Therefore we derive for every u ∈ F εE which implies that Q k is continuous for every k ∈ N because the finite set {ζ K j 1 ≤ j ≤ k} ⊂ F ′ is equicontinuous. The space F εE is barrelled in both cases by Remark 3.10, Q k converges to id in L σ (F εE, F εE) and hence the sequence (Q k ) is equicontinuous by the uniform boundedness principle. Thus F εE has BAP.

Remark.
a) The condition that X is a semi-reflexive, metrisable locally convex space already implies that X is a reflexive Fréchet space by [10, 11.4.3 Proposition, p. 228]. b) The version of Theorem 4.17 for X ′ b still remains valid if we assume that Ψ is antilinear and replace (17) by Indeed, with the notion ζ K n ∶ f ↦ ζ K n (f ) we also have under our new assumptions which is all we need. However, we do not know how to adapt our proof of Theorem 4.17 for F under the new assumptions since Ψ and ζ K n need not be linear if K = C. Our first application of the duality method is quite simple, namely, the already mentioned Fourier expansion in separable Hilbert spaces. Let (H, ⟨⋅, ⋅⟩) be a real, infinite dimensional Hilbert space with orthonormal Schauder basis (f n ). Then the associated coefficient functionals are given by (⟨⋅, f n ⟩) and due to the Riesz representation theorem we have H ≅ H ′ b via the linear (K = R) map Ψ∶ x ↦ ⟨⋅, x⟩. Therefore ⟨x, f n ⟩ = ⟨f n , x⟩ = Ψ(x)(f n ) for all x ∈ H and n ∈ N because H is a real space, and we obtain by where the series converge in L b (H, E) resp. HεE. b) If E is complete (resp. quasi-, sequentially complete), then ε E and HεE ≅ H⊗ * ε E with * = c (resp. qc, sc). Further, where the series converge in H ′ b⊗ * ε E resp. H⊗ * ε E. c) If E is a semi-Montel (resp. complete semi-Montel) space, then with * = qc (resp. c) and where the series converges in L b (H, E). Let us turn to the sequence spaces ℓ p ∶= {x ∈ K N x p ∶= (∑ ∞ n=1 x n p ) 1 p < ∞} for 1 ≤ p < ∞. They are Banach spaces, the unit sequences (ϕ n ) n∈N from Proposition 4.3 form a Schauder basis of the ℓ p spaces for every 1 ≤ p < ∞ and the associated coefficient functionals are (δ n ). Let 1 < p, q < ∞ with 1 p + 1 q = 1. The isomorphy ℓ p ≅ (ℓ q ) ′ b is given by the map x n y n , and the same mapping rule yields the isomorphy ℓ 1 ≅ c 0 (N) ′ b . We observe that for 1 ≤ p < ∞ δ n (x) = x n = Ψ(x)(ϕ n ), x ∈ ℓ p , n ∈ N, and hence we get from Theorem 4.17: 4.20. Corollary. Let E be an lcHs and 1 < p, q < ∞ with 1 p + 1 q = 1. a) (ℓ q ) ′ b ⊗ E resp. ℓ p ⊗ E is sequentially dense in (ℓ q ) ′ b εE resp. ℓ p εE and where the series converge in L b (ℓ q , E) resp. ℓ p εE. b) If E is complete (resp. quasi-, sequentially complete), then (ℓ q ) ′ b εE ≅ (ℓ q ) ′ b⊗ * ε E and ℓ p εE ≅ ℓ p⊗ * ε E with * = c (resp. qc, sc). Further, where the series converge in (ℓ q ) ′ b⊗ * ε E resp. ℓ p⊗ * ε E. c) If E is a semi-Montel (resp. complete semi-Montel) space, then b⊗ * ε E with * = qc (resp. c) and where the series converges in L b (ℓ q , E). d) If E is a Fréchet space, then (ℓ q ) ′ b εE and ℓ p εE have BAP. If p = q = 2, we see that the replacement of the map x ↦ ⟨⋅, x⟩ by Ψ repaired the defect of Corollary 4.19 being restricted to K = R.