DOCSLIB.ORG

Series Representations in Spaces of Vector‐Valued Functions Via

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Series Representations in Spaces of Vector‐Valued Functions Via 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 =.
Load more

Recommended publications
  • Arxiv:Math/0605475V1 [Math.FA] 17 May 2006 Td Lse Fsprbebnc Pcsi B2 O] Eproved He Bo3]
    SOME STRONGLY BOUNDED CLASSES OF BANACH SPACES PANDELIS DODOS AND VALENTIN FERENCZI Abstract. We show that the classes of separable reflexive Banach spaces and of spaces with separable dual are strongly bounded. This gives a new proof of a recent result of E. Odell and Th. Schlumprecht, asserting that there exists a separable reflexive Ba- nach space containing isomorphic copies of every separable uni- formly convex Banach spaces. 1. Introduction A Banach space X is said to be universal for a class C of Banach spaces when every space in C embeds isomorphically into X. It is complementably universal when the embeddings are complemented. According to the classical Mazur Theorem, the space C(2N) is uni- versal for separable Banach spaces. By [JS], there does not exist a sep- arable Banach space which is complementably universal for the class of separable Banach spaces. However, A. Pe lczy´nski [P] constructed a space U with a Schauder basis which is complementably universal for the class of spaces with a Schauder basis (and even for the class of spaces with the Bounded Approximation Property). There is also an unconditional version of U, i.e. a space with an unconditional ba- sis which is complementably universal for the class of spaces with an unconditional basis. In 1968, W. Szlenk proved that there does not exist a Banach space arXiv:math/0605475v1 [math.FA] 17 May 2006 with separable dual which is universal for the class of reflexive Banach spaces [Sz]. His proof is based on the definition of the Szlenk index which is a transfinite measure of the separability of the dual of a space.
    [Show full text]
  • On the Hypercyclicity Criterion for Operators of Read's Type
    ON THE HYPERCYCLICITY CRITERION FOR OPERATORS OF READ’S TYPE by Sophie Grivaux Abstract. — Let T be a so-called operator of Read’s type on a (real or complex) separable Banach space, having no non-trivial invariant subset. We prove in this note that T ⊕ T is then hypercyclic, i.e. that T satisfies the Hypercyclicity Criterion. 1. The Invariant Subspace Problem Given a (real or complex) infinite-dimensional separable Banach space X, the Invariant Subspace Problem for X asks whether every bounded operator T on X admits a non- trivial invariant subspace, i.e. a closed subspace M of X with M 6= {0} and M 6= X such that T (M) ⊆ M. It was answered in the negative in the 80’s, first by Enflo [11] and then by Read [24], who constructed examples of separable Banach spaces supporting operators without non-trivial closed invariant subspace. One of the most famous open questions in modern operator theory is the Hilbertian version of the Invariant Subspace Problem, but it is also widely open in the reflexive setting: to this day, all the known examples of operators without non-trivial invariant subspace live on non-reflexive Banach spaces. Read provided several classes of operators on ℓ1(N) having no non-trivial invariant subspace [25], [26], [29]. In the work [28], he gave examples of such operators on c0(N) and X = Lℓ2 J, the ℓ2-sum of countably many copies of the James space J; since J is quasi reflexive (i.e. has codimension 1 in its bidual J ∗∗), the space X has the property that X∗∗/X is separable.
    [Show full text]
  • A Representation Theorem for Schauder Bases in Hilbert Space
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 2, February 1998, Pages 553{560 S 0002-9939(98)04168-9 A REPRESENTATION THEOREM FOR SCHAUDER BASES IN HILBERT SPACE STEPHANE JAFFARD AND ROBERT M. YOUNG (Communicated by Dale Alspach) Abstract. A sequence of vectors f1;f2;f3;::: in a separable Hilbert space f g H is said to be a Schauder basis for H if every element f H has a unique norm-convergent expansion 2 f = cnfn: If, in addition, there exist positive constantsX A and B such that 2 2 2 A cn cnfn B cn ; j j ≤ ≤ j j then we call f1;f2;fX3;::: a Riesz X basis. In theX first half of this paper, we f g show that every Schauder basis for H can be obtained from an orthonormal basis by means of a (possibly unbounded) one-to-one positive self adjoint oper- ator. In the second half, we use this result to extend and clarify a remarkable theorem due to Duffin and Eachus characterizing the class of Riesz bases in Hilbert space. 1. The main theorem A sequence of vectors f1;f2;f3;::: in a separable Hilbert space H is said to be a Schauder basis for H{ if every element} f H has a unique norm-convergent expansion ∈ f = cnfn: If, in addition, there exist positive constantsX A and B such that 2 A c 2 c f B c 2; | n| ≤ n n ≤ | n| then we call f1;f2;f3;:::X a Riesz basis.X Riesz basesX have been studied intensively ever since Paley{ and Wiener} first recognized the possibility of nonharmonic Fourier expansions iλnt f = cne for functions f in L2( π, π) (see, for example,X [2] and [5] and the references therein).
    [Show full text]
  • Pre-Schauder Bases in Topological Vector Spaces
    S S symmetry Article Pre-Schauder Bases in Topological Vector Spaces Francisco Javier García-Pacheco *,† and Francisco Javier Pérez-Fernández † Department of Mathematics, University of Cadiz, 11519 Puerto Real, Spain * Correspondence: [email protected] † These authors contributed equally to this work. Received: 17 June 2019; Accepted: 2 August 2019; Published: 9 August 2019 Abstract: A Schauder basis in a real or complex Banach space X is a sequence (en)n2N in X such ¥ that for every x 2 X there exists a unique sequence of scalars (ln)n2N satisfying that x = ∑n=1 lnen. Schauder bases were first introduced in the setting of real or complex Banach spaces but they have been transported to the scope of real or complex Hausdorff locally convex topological vector spaces. In this manuscript, we extend them to the setting of topological vector spaces over an absolutely valued division ring by redefining them as pre-Schauder bases. We first prove that, if a topological vector space admits a pre-Schauder basis, then the linear span of the basis is Hausdorff and the series linear span of the basis minus the linear span contains the intersection of all neighborhoods of 0. As a consequence, we conclude that the coefficient functionals are continuous if and only if the canonical projections are also continuous (this is a trivial fact in normed spaces but not in topological vector spaces). We also prove that, if a Hausdorff topological vector space admits a pre-Schauder basis and is w∗-strongly torsionless, then the biorthogonal system formed by the basis and its coefficient functionals is total.
    [Show full text]
  • Functional Analysis Homework 4
    Functional Analysis Homework 4 Due Tuesday, 13 February 2018 Problem 1. A sequence (en) of elements from a normed space X is said to be a Schauder basis if for every Pn x 2 X, there is a unique sequence (cn) of scalars such that i=1 cnen ! x. Show that if a normed space has a Schauder basis, then it is separable.[1] Proof. Suppose X is a normed space with a Schauder basis (en). Note that the scalar field F of X has a countable dense set which we will call G (for R it is Q, and for C it is Q + iQ). Let Y be the collection of all finite linear combinations of elements of the Schauder basis with coefficients in G; that is, ( n ) [ X Y := Y ; where Y := c e c 2 = span fe ; : : : ; e g: n n i i i G G 1 n n2N i=1 [2] n Note that Yn is countable because there is a natural bijective map G ! Yn given by (c1; : : : ; cn) ! Pn i=1 ciei. Since Y is a countable union of countable sets, it is countable. Now we claim that Y is dense in X. To this end, take any x 2 X. Since (en) is a Schauder basis, there Pn exists a sequence of scalars (cn) from F such that i=1 ciei ! x. Therefore, given " > 0, there is some N 2 N such that N X " c e − x < : i i 2 i=1 " Moreover, since is dense in , for each 1 ≤ i ≤ N, there is some ai 2 such that jai − cij ≤ .
    [Show full text]
  • Lecture Slides (Pdf)
    When James met Schreier Niels Jakob Laustsen (Joint work with Alistair Bird) Lancaster University England Bedlewo, July 2009 1 Outline We amalgamate two important classical examples of Banach spaces: I James’ quasi-reflexive Banach spaces, and I Schreier’s space giving a counterexample to the Banach–Saks property, to obtain a family of James–Schreier spaces. We then investigate their properties. Key point: like the James spaces, each James–Schreier space is a commutative Banach algebra with a bounded approximate identity when equipped with the pointwise product. Note: this is joint work with Alistair Bird who will cover further aspects of it in his talk. 2 James’ quasi-reflexive Banach space J — motivation I Defined by James in 1950–51. I Key property: J is quasi-reflexive: dim J∗∗/κ(J) = 1; where κ: J ! J∗∗ is the canonical embedding. I Resolved two major open problems: I a Banach space with separable bidual need not be reflexive; I a separable Banach space which is (isometrically) isomorphic to its bidual need not be reflexive. I Subsequently, many other interesting properties have been added to this list, for instance: I Bessaga & Pełczyński (1960): An infinite-dimensional Banach space X need not be isomorphic to its Cartesian square X ⊕ X . I Edelstein–Mityagin (1970): There may be characters on the Banach algebra B(X ) of bounded linear operators on an infinite-dimensional Banach space X . Specifically, the quasi-reflexivity of J implies that dim B(J)=W (J) = 1; where W (J) is the ideal of weakly compact operators on J. 3 Conventions I Throughout, the scalar field is either K := R or K := C.
    [Show full text]
  • NORMALIZATION PROPERTIES of SCHAUDER BASES by N
    NORMALIZATION PROPERTIES OF SCHAUDER BASES By N. J. KALTON [Received 21 May 1969] 1. Introduction A basis of a locally convex topological vector space E is a sequence (xn) such that for every x e E there is a unique sequence (<xn) of scalars such 00 x that x = 2 &n n > if> further, the linear functionals fn{x) = ixn are con- n=l tinuous then (xn) is said to be a Schauder basis. The theory of Schauder bases has been developed in considerable detail in Banach spaces, but comparatively little is known about bases of more general types of spaces (see [1] and [2]). When E is a normed space, an alternative Schauder basis is given by yn = aJ7l/||a?J|; this basis is normalized so that \\yn\\ = 1. In topological terms this means that (yn) is bounded and 'bounded away from zero' in the sense that there exists a neighbourhood V of 0 such that yn $ V for all n. These two properties are extremely useful in the theory of bases in Banach spaces. The purpose of this paper is to consider these properties for bases of general locally convex spaces. DEFINITION 1.1. A sequence (xn) is said to be regular if there exists a neighbourhood V of 0 such that xn $ V for all n. Even if E is a Frechet space, it does not necessarily follow that a basis (xn) can be 'regularized'. If o» denotes the space of all real (or complex) sequences, and cp is the space of all real sequences eventually equal to zero, then a> and <p form a dual pair of sequence spaces and (OJ, a(oj,<p)) is an .F-space.
    [Show full text]
  • Schauder Equivalence Relations BLAST2015, University of North Texas
    Schauder equivalence relations BLAST2015, University of North Texas Xin Ma Department of Mathematics, Texas A&M University Xin Ma (Texas A&M University) June. 12, 2015 1 / 21 Plan of my talk 1 Introduction 2 Reducibility and non-reducibility 3 Structure theorems Xin Ma (Texas A&M University) June. 12, 2015 2 / 21 Now, Let X ,Y be Polish spaces and E,F be equivalence relations on X and Y , respectively. Definition a Borel reduction from E to F , denoted by E ≤B F , is a Borel function θ : X ! Y such that for all x; y 2 X , xEy () θ(x)F θ(y). We write E ∼B F if both E ≤B F and F ≤B E hold. We also write E <B F if E ≤B F but not F ≤B E . Xin Ma (Texas A&M University) June. 12, 2015 3 / 21 Banach space and Schauder equivalence relations Definition Given a Banach space X , a Schauder basis fxng of X means that any 1 P x 2 X can be expanded by the form x = anxn for a unique fang in RN. n=1 A sequence fxng is a basis of its closed linear span is called a basic sequence. Any separable Banach space is a Polish space, we consider Schauder equivalence relations defined as follows. This definition is due to Ding. Definition (Ding, 2013) For fxng is a basic sequence of Banach space X , we denote coef (X ; (xn)) P to be the set of all a = (an) 2 RN such that anxn converges. Define the equivalence relation E(X ; (xn)) by, for any x; y 2 RN, (x; y) 2 E(X ; (xn)) () x − y 2 coef (X ; (xn)): Xin Ma (Texas A&M University) June.
    [Show full text]
  • Invariant Subspaces of Compact Operators on Topological Vector Spaces
    Pacific Journal of Mathematics INVARIANT SUBSPACES OF COMPACT OPERATORS ON TOPOLOGICAL VECTOR SPACES ARTHUR D. GRAINGER Vol. 56, No. 2 December 1975 PACIFIC JOURNAL OF MATHEMATICS Vol. 56. No. 2, 1975 INVARIANT SUBSPACES OF COMPACT OPERATORS ON TOPOLOGICAL VECTOR SPACES ARTHUR D. GRAINGER Let (//, r) be a topological vector space and let T be a compact linear operator mapping H into H (i.e., T[V] is contained in a r- compact set for some r- neighborhood V of the zero vector in H). Sufficient conditions are given for (H,τ) so that T has a non-trivial, closed invariant linear subspace. In particular, it is shown that any complete, metrizable topological vector space with a Schauder basis satisfies the conditions stated in this paper. The proofs and conditions are stated within the framework of nonstandard analysis. Introduction. This paper considers the following problem: given a compact operator T (Definition 2.11) on a topological vector space (H, r), does there exist a closed nontrivial linear subspace F of H such that T[F] CF? Aronszajn and Smith gave an affirmative answer to the above question when H is a Banach space (see [1]). Also it is easily shown that the Aronszajn and Smith result can be extended to locally convex spaces. However, it appears that other methods must be used for nonlocally convex spaces. Sufficient conditions are given for a topological vector space so that a compact linear operator defined on the space has at least one nontrivial closed invariant linear subspace (Definitions 2.1 and 4.1, Theorems 3.2, 4.2 and 4.7).
    [Show full text]
  • Some Open Problems in Banach Space Theory Séminaire Choquet
    Séminaire Choquet. Initiation à l’analyse JORAM LINDENSTRAUSS Some open problems in Banach space theory Séminaire Choquet. Initiation à l’analyse, tome 15 (1975-1976), exp. no 18, p. 1-9 <http://www.numdam.org/item?id=SC_1975-1976__15__A6_0> © Séminaire Choquet. Initiation à l’analyse (Secrétariat mathématique, Paris), 1975-1976, tous droits réservés. L’accès aux archives de la collection « Séminaire Choquet. Initiation à l’analyse » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Seminaire CHOQUET 18-01 (Initiation a l’analyse) anné e I- t975/76, nO’ 18 ,, 91 p~ 8 avril 1976 SOME OPEN PROBLEMS IN BANACH SPACE THEORY by Joram LINDENSTRAUSS My aimnn this talk is to discuss briefly some directions of research; in the iso- morphic theory of Banach spaces. The’directions which I plan to discuss are the following. ’ (a) Existence of operators, Local of Banach . (b) theory spaces, , (c) The approximation property , (d) Existence of bases, (e). Non linear problems, (f) Nbn separable problems. In each of these directions, I shall discuss only very few open questions so as to illustrate the specific direction. It should be emphasized that the choice of these is questions based on my personal interest and there are many other (often more fun- damental) open questions in these directions which are currently being studied by various mathematicians.
    [Show full text]
  • On a Semigroup Problem II
    S S symmetry Article On a Semigroup Problem II Viorel Nitica 1,2,* and Andrew Torok 2,3 1 Department of Mathematics, West Chester University of Pennsylvania, West Chester, PA 19383, USA 2 Institute of Mathematics, Romanian Academy, P.O. Box 1–764, RO-70700 Bucharest, Romania; [email protected] 3 Department of Mathematics, University of Houston, Houston, TX 77204-3308, USA * Correspondence: [email protected] Received: 6 April 2020; Accepted: 19 August 2020; Published: 21 August 2020 Abstract: We consider the following semigroup problem: is the closure of a semigroup S in a topological vector space X a group when S does not lie on “one side” of any closed hyperplane of X? Whereas for finite dimensional spaces, the answer is positive, we give a new example of infinite dimensional spaces where the answer is negative. Keywords: semigroup; group; linear topological space; separation by functionals 1. Introduction We consider the following question. Semigroup problem: Let X be a topological vector space (TVS) over the real field R, and let S ⊂ X be a semigroup (By a semigroup in a vector space X, we mean a subset S ⊂ X that is closed under addition. It does not have to contain the zero vector). Let X0 be the topological closure of the linear span of S (that is, the linear span of S is dense in X0, and the complement of X0 is open). Assume that ∗ S is not separated by any continuous linear functional in the (real) dual of X0: for any f 2 (X0) n f0g, there are x1, x2 2 S such that f(x1) > 0 and f(x2) < 0.
    [Show full text]
  • Arxiv:1806.01889V5 [Math.FA]
    SERIES REPRESENTATIONS IN SPACES OF VECTOR-VALUED FUNCTIONS VIA SCHAUDER DECOMPOSITIONS KARSTEN KRUSE Abstract. It is a classical result that every 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 C. Motivated by this example we try to answer the following question. Let E be a locally convex Hausdorff space over a field K, F(Ω) be a locally convex Hausdorff space of K-valued functions on a set Ω and F(Ω,E) be an E-valued counterpart of F(Ω) (where the term E-valued counterpart needs clarification itself). For which spaces is it possible to lift series representations of elements of F(Ω) to elements of F(Ω,E)? We derive sufficient conditions for the answer to be affirmative using Schauder decompositions which are applicable for many classical spaces of functions F(Ω) having an equicontinuous Schauder basis. 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 Dr ⊂ C be an open disc around zero with radius r > 0 and O(Dr) be the space of holomorphic functions on Dr, i.e. the space of functions f∶ Dr → C such that the limit (1) f(z + h) − f(z) D f (z) ∶= lim , z ∈ r, (1) h→0 h h∈C,h≠0 C D exists in .
    [Show full text]