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Valued Functions Via Schauder Decompositions

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Valued Functions Via Schauder Decompositions Received: 14 April 2019 Revised: 24 October 2019 Accepted: 13 December 2019 DOI: 10.1002/mana.201900172 ORIGINAL PAPER Series representations in spaces of vector-valued functions via Schauder decompositions Karsten Kruse Technische Universität Hamburg, Institut für Mathematik, Am Schwarzenberg- Abstract Campus 3, 21073 Hamburg, Germany It is a classical result that every ℂ-valued holomorphic function has a local power Correspondence series representation. This even remains true for holomorphic functions with val- Karsten Kruse, Technische Universität ues in a locally complete locally convex Hausdorff space over ℂ.Motivatedby Hamburg, Institut für Mathematik, Am this example we try to answer the following question. Let be a locally con- Schwarzenberg-Campus 3, 21073 Ham- (Ω) burg, Germany. vex Hausdorff space over a field ,let be a locally convex Hausdorff space Email: [email protected] of -valued functions on a set Ω and let (Ω, ) be an -valued counterpart of (Ω) (where the term -valued counterpart needs clarification itself). For which spaces is it possible to lift series representations of elements of (Ω) to elements of (Ω, )? We derive sufficient conditions for the answer to be affirmative using Schauder decompositions which are applicable for many classical spaces of func- tions (Ω) having an equicontinuous Schauder basis. KEYWORDS injective tensor product, Schauder basis, Schauder decomposition, series representation, vector-valued function MSC (2020) Primary: 46E40; Secondary: 46A32; 46E10 1 INTRODUCTION The purpose of this paper is to lift series representations known from scalar-valued functions to vector-valued functions and its underlying idea was derived from the classical example of the (local) power( ) series representation of a holomorphic function. Let ⊂ℂbe an open disc around zero with radius >0and let be the space of holomorphic functions on , i.e. the space of functions ∶ →ℂsuch that the limit (1) (+ℎ)−() () ∶= lim ,∈, (1.1) ℎ→0 ℎ ℎ∈ℂ,ℎ≠0 This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. © 2020 The Authors. Mathematische Nachrichten published by Wiley-VCH GmbH 354 www.mn-journal.org Mathematische Nachrichten. 2021;294:354–376. KRUSE 355 ( ) exists in ℂ. It is well-known that every ∈ can be written as ∑∞ ()(0) () = ,∈, ! =0 () where the power series on the right-hand side converges uniformly on every compact subset of and (0) is the -th complex derivative of at 0 which is defined from (1.1) by the recursion ( )(1) (0) ∶= and () ∶= (−1) for ∈ℕ. By [9, 2.1 Theorem and Definition, p. 17–18] and [9, 5.2 Theorem, p. 35], this series representation remains valid if is a holomorphic function on with values in a locally complete locally convex Hausdorff space over ℂ where holomorphy means that the limit (1.1) exists( ) in and the higher complex derivatives are defined recursively as well. Analysing this example, we observe that , equipped with the topology of uniform convergence on compact subsets of ,isa Fréchet space, in particular barrelled, with a Schauder basis formed by the monomials ↦. Further, the formulas for the complex derivatives of a ℂ-valued resp. an -valued function on are built up in the same way by (1.1). 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 [11, 14.2, p. 292] that a sequence in a locally convex Hausdorff space (over a) field is called a topological basis, or simply a basis, if for every ∈there is a unique sequence of coefficients () in such that ∑∞ = () (1.2) =1 where the series converges in . Due to the uniqueness( of the) coefficients the map ∶↦ () is well-defined, linear and called the -th coefficient functional associated to . Further, for each ∈ℕthe map ∑ ∶→,() ∶= (), =1 ( ) ( ) is a linear( ) projection whose range is span 1,…, anditiscalledthe-th expansion operator associated to .A basis of is called equicontinuous if the expansion operators form an equicontinuous( ) sequence in the linear space (,) of continuous linear maps from to (see [11,14.3,p.296]).Abasis of is called a Schauder basis if ′ the coefficient functionals are continuous, i.e. ∈ ∶= (, ) for each ∈ℕ. In particular, this is already fulfilled if is a Fréchet space by [18, Corollary 28.11, p. 351]. If is barrelled, then a Schauder basis of is already equicontinuous and has the (bounded) approximation property by the uniform boundedness principle. The starting point for our approach is Equation( ) (1.2). Let and be locally convex Hausdorff( ) spaces over a field where( ) has an equicontinuous Schauder basis with associated coefficient functionals . The expansion operators form a so-called Schauder decomposition of (see [4, p. 77]), i.e. they are continuous projections on such that (i) =min(,) for all , , ∈ℕ (ii) ( ≠) for ≠ , (iii) converges to for each ∈. This operator theoretic definition of a Schauder decomposition is equivalent to the usual definition( in) terms of closed subspaces of given in [14,p.377](see[17, p. 219]).( In) our main Theorem 3.1 we prove that id is a Schauder ′ decomposition of Schwartz’ -product ∶= , and each ∈has the series representation ∑∞ ( ) ( ) ′ ′ ′ ′ ( )= ,∈. =1 356 KRUSE Now, suppose that =(Ω) is a space of -valued functions on a set Ω with a topology such that the point-evaluation functionals , ∈Ω, are continuous and that there is a locally convex Hausdorff space (Ω, ) of functions from Ω to such that the map [ ( )] ∶(Ω) → (Ω, ), ⟼ ↦ , is a (linear topological) isomorphism. Assuming that for each ∈ℕand ∈(Ω) there is (()) ∈ with ( ) (()) = , (1.3) ( ( ) ) ◦ id ◦−1 (Ω, ) we obtain in Corollary 3.6 that is a Schauder decomposition of and ( ( ) ) ∑∞ −1 = lim ◦ id ◦ () = (),∈(Ω, ), →∞ =1 which is the desired series representation in (Ω, ). Condition (1.3) might seem strange at a first glance but for example in the case of -valued( holomorphic) functions on( it) guarantees that( the) complex derivatives at 0 appear in the Schauder () () () decomposition of , since () (0) = 0 for all ∈ and ∈ℕ0 where 0 is the point-evaluation of the -th complex derivative. We apply our result to sequence spaces, spaces of continuously differentiable functions on a compact interval, the space of holomorphic functions, the Schwartz space and the space of smooth functions which are 2-periodic in each variable. ˆ As a byproduct of Theorem 3.1 we obtain that every element of the completion ⊗ of the injective tensor product has a series representation as well if is a complete space with an equicontinuous Schauder basis and is complete. ˆ ˆ Concerning series representation in ⊗, little seems to be known whereas for the completion ⊗ of the projective ˆ tensor product ⊗ of two metrisable locally convex spaces and it is well-known that every ∈⊗ has a series representation ∑∞ = ⊗ =1 ( ) ( ) ( ) ( ) 1 where ∈ ,i.e. is absolutely summable, and and are null sequences in and , respectively (see e.g. [10, Chap. I, §2 , n◦1, Théorème 1, p. 51] or [11, 15.6.4 Corollary, p. 334]). If and are metrisable and one of them ˆ ˆ is nuclear, then the isomorphism ⊗≅⊗ holds and we trivially have a series representation of the elements of ˆ ˆ ⊗ as well. Other conditions on the existence of series representations of the elements of ⊗ canbefoundin[19, Proposition 4.25, p. 88], where and are Banach spaces and both of them have a Schauder basis, and in [12, Theorem 2, p. 283], where and are locally convex Hausdorff spaces and both of them have an equicontinuous Schauder basis. 2 NOTATION AND PRELIMINARIES We equip the spaces ℝ, ∈ℕ,andℂ with the usual Euclidean norm | ⋅ |. For a subset of a topological vector space , we write acx() for the closure of the absolutely convex hull acx() of in . By we always denote a non-trivial locally convex( Hausdorff) space, in short lcHs,( over) the field =ℝor ℂ equipped = ∶= {| ⋅ |} with a directed fundamental system of seminorms ∈.If , then we⋃ set ∈ . We recall that for ⊂ ∶= adisk , i.e. a bounded, absolutely convex set, the vector space ∈ℕ becomes a normed space if it is equipped with the gauge functional of as a norm (see [11, p. 151]). The space is called locally complete if is a Banach space for every closed disk ⊂(see [11, 10.2.1 Proposition, p. 197]). For more details on the theory of locally convex spaces see [8, 11]or[18]. Ω By we denote the set of maps from a non-empty set Ω to a non-empty set ,by the characteristic function of a subset ⊂Ωand by (, ) the space of continuous linear operators from to where and are locally convex Haus- dorff spaces. If =,wejustwrite′ ∶= (, ) for the dual space. If and are (linearly topologically) isomorphic, we write ≅and, if is only isomorphic to a subspace of ,wewrite↪̃ . We denote by (, ) the space (, ) KRUSE 357 equipped with the locally convex topology of uniform convergence on the finite subsets of if =, on the absolutely convex compact subsets of if =and on the precompact (totally bounded) subsets of if =. The so-called -product of Schwartz is defined by ( ) ′ ∶= , (2.1) ( ) ′ ′ where , is equipped with the topology of uniform convergence on equicontinuous subsets of . This definition of the -product coincides with the original one by Schwartz [23, Chap. I, §1, Définition, p. 18]. It is symmetric which means that ≅. In the literature the definition of the -product is sometimes done the other way around, i.e. is defined by the right-hand side of (2.1) but due to the symmetry these definitions are equivalent and for our purpose ′ ′ the given definition is more suitable. If we replace by , we obtain Grothendieck’s definition of the -product and ′ ′ we remark that the two -products coincide if is quasi-complete because then =.
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