Topological Characterization of Weakly Compact Operators Revisited †
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The Uniform Boundedness Theorem in Asymmetric Normed Spaces
Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2012, Article ID 809626, 8 pages doi:10.1155/2012/809626 Research Article The Uniform Boundedness Theorem in Asymmetric Normed Spaces C. Alegre,1 S. Romaguera,1 and P. Veeramani2 1 Instituto Universitario de Matematica´ Pura y Aplicada, Universitat Politecnica` de Valencia,` 46022 Valencia, Spain 2 Department of Mathematics, Indian Institute of Technology Madras, Chennai 6000 36, India Correspondence should be addressed to C. Alegre, [email protected] Received 5 July 2012; Accepted 27 August 2012 Academic Editor: Yong Zhou Copyright q 2012 C. Alegre et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We obtain a uniform boundedness type theorem in the frame of asymmetric normed spaces. The classical result for normed spaces follows as a particular case. 1. Introduction Throughout this paper the letters R and R will denote the set of real numbers and the set of nonnegative real numbers, respectively. The book of Aliprantis and Border 1 provides a good basic reference for functional analysis in our context. Let X be a real vector space. A function p : X → R is said to be an asymmetric norm on X 2, 3 if for all x, y ∈ X,andr ∈ R , i pxp−x0 if and only if x 0; ii prxrpx; iii px y ≤ pxpy. The pair X, p is called an asymmetric normed space. Asymmetric norms are also called quasinorms in 4–6, and nonsymmetric norms in 7. -
Notes on Linear Functional Analysis by M.A Sofi
Linear Functional Analysis Prof. M. A. Sofi Department of Mathematics University of Kashmir Srinagar-190006 1 Bounded linear operators In this section, we shall characterize continuity of linear operators acting between normed spaces. It turns out that a linear operator is continuous on a normed linear space as soon as it is continuous at the origin or for that matter, at any point of the domain of its definition. Theorem 1.1. Given normed spaces X and Y and a linear map T : X ! Y; then T is continuous on X if and only if 9 c > 0 such that kT (x)k ≤ ckxk; 8 x 2 X: (1) Proof. Assume that T is continuous. In particular, T is continuous at the origin. By the definition of continuity, since T (0) = 0; there exits a neigh- bourhood U at the origin in X such that T (U) ⊆ B(Y ): But then 9r > 0 such that Sr(0) ⊂ U: This gives T (Sr(0)) ⊆ T (U) ⊆ B(Y ): (2) x Let x(6= 0) 2 X; for otherwise, (1) is trivially satisfied. Then kxk 2 Sr(0); so rx that by (2), T 2kxk 2 B(Y ): In other words, 2 kT (x)k ≤ ckxk where c = r which gives (1). Conversely, assume that (1) is true. To show that T is continuous on X; let 1 n n x 2 X and assume that xn −! x in X: Then xn − x −! 0 in X: Thus, 8 > 0; 9 N 3: 8 n > N; kT (xn) − T (x)k = kT (xn − x)k ≤ ckxn − xk ≤ , n which holds for all n > N: In other words, T (xn) −! T (x) in Y and therefore, T is continuous at x 2 X: Since x 2 X was arbitrarily chosen, it follows that T is continuous on X: Definition 1.2. -
Eberlein-Šmulian Theorem and Some of Its Applications
Eberlein-Šmulian theorem and some of its applications Kristina Qarri Supervisors Trond Abrahamsen Associate professor, PhD University of Agder Norway Olav Nygaard Professor, PhD University of Agder Norway This master’s thesis is carried out as a part of the education at the University of Agder and is therefore approved as a part of this education. However, this does not imply that the University answers for the methods that are used or the conclusions that are drawn. University of Agder, 2014 Faculty of Engineering and Science Department of Mathematics Contents Abstract 1 1 Introduction 2 1.1 Notation and terminology . 4 1.2 Cornerstones in Functional Analysis . 4 2 Basics of weak and weak* topologies 6 2.1 The weak topology . 7 2.2 Weak* topology . 16 3 Schauder Basis Theory 21 3.1 First Properties . 21 3.2 Constructing basic sequences . 37 4 Proof of the Eberlein Šmulian theorem due to Whitley 50 5 The weak topology and the topology of pointwise convergence on C(K) 58 6 A generalization of the Ebrlein-Šmulian theorem 64 7 Some applications to Tauberian operator theory 69 Summary 73 i Abstract The thesis is about Eberlein-Šmulian and some its applications. The goal is to investigate and explain different proofs of the Eberlein-Šmulian theorem. First we introduce the general theory of weak and weak* topology defined on a normed space X. Next we present the definition of a basis and a Schauder basis of a given Banach space. We give some examples and prove the main theorems which are needed to enjoy the proof of the Eberlein-Šmulian theorem given by Pelchynski in 1964. -
Norm Attaining Functionals on C(T)
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 1, January 1998, Pages 153{157 S 0002-9939(98)04008-8 NORM ATTAINING FUNCTIONALS ON C(T ) P. S. KENDEROV, W. B. MOORS, AND SCOTT SCIFFER (Communicated by Dale Alspach) Abstract. It is shown that for any infinite compact Hausdorff space T ,the Bishop-Phelps set in C(T )∗ is of the first Baire category when C(T )hasthe supremum norm. For a Banach space X with closed unit ball B(X), the Bishop-Phelps set is the set of all functionals in the dual X∗ which attain their supremum on B(X); that is, the set µ X∗:thereexistsx B(X) with µ(x)= µ .Itiswellknownthat { ∈ ∈ k k} the Bishop-Phelps set is always dense in the dual X∗. It is more difficult, however, to decide whether the Bishop-Phelps set is residual as this would require a concrete characterisation of those linear functionals in X∗ which attain their supremum on B(X). In general it is difficult to characterise such functionals, however it is possible when X = C(T ), for some compact T , with the supremum norm. The aim of this note is to show that in this case the Bishop-Phelps set is a first Baire category subset of C(T )∗. A problem of considerable interest in differentiability theory is the following. If the Bishop-Phelps set is a residual subset of the dual X ∗, is the dual norm necessarily densely Fr´echet differentiable? It is known, for example, that this is the case if the dual X∗ is weak Asplund, [1, Corollary 1.6(i)], or if X has an equivalent weakly mid-point locally uniformly rotund norm and the weak topology on X is σ-fragmented by the norm, [2, Theo- rems 3.3 and 4.4]. -
A Dual Differentiation Space Without an Equivalent Locally Uniformly Rotund Norm
J. Aust. Math. Soc. 77 (2004), 357-364 A DUAL DIFFERENTIATION SPACE WITHOUT AN EQUIVALENT LOCALLY UNIFORMLY ROTUND NORM PETAR S. KENDEROV and WARREN B. MOORS (Received 23 August 2002; revised 27 August 2003) Communicated by A. J. Pryde Abstract A Banach space (X, || • ||) is said to be a dual differentiation space if every continuous convex function defined on a non-empty open convex subset A of X* that possesses weak* continuous subgradients at the points of a residual subset of A is Frechet differentiable on a dense subset of A. In this paper we show that if we assume the continuum hypothesis then there exists a dual differentiation space that does not admit an equivalent locally uniformly rotund norm. 2000 Mathematics subject classification: primary 46B20; secondary 58C20. Keywords and phrases: locally uniformly rotund norm, Bishop-Phelps set, Radon-Nikodym property. 1. Introduction Given a Banach space (X, || • ||) the Bishop-Phelps set (or BP-set for short) is the set {x* € X* : ||**|| =**(*) for some x e Bx], where Bx denotes the closed unit ball in (X, || • ||). The Bishop-Phelps theorem, [1] says that the BP-set is always dense in X*. In this paper we are interested in the case when the BP-set is residual (that is, contains a dense Gs subset) in X*. Certainly, it is known that if the dual norm is Frechet differentiable on a dense subset of X* then the BP-set is residual in X* (see the discussion in [13]). However, the converse question (that is, if the BP-set is residual in X* must the dual norm necessarily be Frechet differentiable on a dense subset of X*?) remains open. -
Locally Convex Spaces and Schur Type Properties
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 44, 2019, 363–378 LOCALLY CONVEX SPACES AND SCHUR TYPE PROPERTIES Saak Gabriyelyan Ben-Gurion University of the Negev, Department of Mathematics Beer-Sheva, P.O. 653, Israel; [email protected] Abstract. In the main result of the paper we extend Rosenthal’s characterization of Banach spaces with the Schur property by showing that for a quasi-complete locally convex space E whose separable bounded sets are metrizable the following conditions are equivalent: (1) E has the Schur property, (2) E and Ew have the same sequentially compact sets, where Ew is the space E with the weak topology, (3) E and Ew have the same compact sets, (4) E and Ew have the same countably compact sets, (5) E and Ew have the same pseudocompact sets, (6) E and Ew have the same functionally bounded sets, (7) every bounded non-precompact sequence in E has a subsequence which is equivalent to the unit basis of ℓ1 and (8) every bounded non-precompact sequence in E has a subsequence which is discrete and C-embedded in Ew. 1. Introduction A locally convex space (lcs for short) E is said to have the Schur property if E and Ew have the same convergent sequences, where Ew is the space E endowed with the ′ weak topology σ(E, E ). By the classical Schur theorem the Banach space ℓ1(Γ) has the Schur property for every set Γ. The results of Rosenthal in [38] (see also §2 of [8]) show that a Banach space E has the Schur property if and only if every δ-separated sequence in the closed unit ball of E has a subsequence equivalent to the unit basis of ℓ1. -
Mackey Continuity of Characteristic Functionals
Georgian Mathematical Journal Volume 9 (2002), Number 1, 83–112 MACKEY CONTINUITY OF CHARACTERISTIC FUNCTIONALS S. KWAPIEN´ AND V. TARIELADZE Dedicated to N. Vakhania on the occassion of his 70th birthday Abstract. Problems of the Mackey-continuity of characteristic functionals and the localization of linear kernels of Radon probability measures in locally convex spaces are investigated. First the class of spaces is described, for which the continuity takes place. Then it is shown that in a non-complete sigma- compact inner product space, as well as in a non-complete sigma-compact metizable nuclear space, there may exist a Radon probability measure having a non-continuous characteristic functional in the Mackey topology and a linear kernel not contained in the initial space. Similar problems for moment forms and higher order kernels are also touched upon. Finally, a new proof of the result due to Chr. Borell is given, which asserts that any Gaussian Radon measure on an arbitrary Hausdorff locally convex space has the Mackey- continuous characteristic functional. 2000 Mathematics Subject Classification: Primary: 60B11. Secondary: 28C20, 60B15. Key words and phrases: Radon probability measure, characteristic func- tional, kernel of a measure, weak topology, Mackey topology, presupport, Gaussian measure, covariance operator. 1. Introduction Recall that for a Hausdorff locally convex space X the Mackey topology ¿(X¤;X) is the topology in its topological dual X¤ of uniform convergence on all weakly compact absolutely convex subsets of X. The aim of this paper is to answer the following three equivalent questions: Question 1. Is the characteristic functional¹ ˆ of a Radon probability mea- sure ¹ in a Hausdorff locally convex space X continuous in the Mackey topology ¿(X¤;X)? Question 2. -
Mackey +0-Barrelled Spaces Stephen A
Advances in Mathematics 145, 230238 (1999) Article ID aima.1998.1815, available online at http:ÂÂwww.idealibrary.com on Mackey +0-Barrelled Spaces Stephen A. Saxon* Department of Mathematics, University of Florida, P.O. Box 118105, Gainesville, Florida 32611-8105 E-mail: saxonÄmath.ufl.edu and Ian Tweddle* Department of Mathematics, University of Strathclyde, Glasgow G11XH, Scotland, United Kingdom E-mail: i.tweddleÄstrath.ac.uk CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - PublisherReceived Connector March 25, 1998; accepted December 14, 1998 In the context of ``Reinventing weak barrelledness,'' the best possible versions of the RobertsonSaxonRobertson Splitting Theorem and the SaxonTweddle Fit and Flat Components Theorem are obtained by weakening the hypothesis from ``barrelled'' to ``Mackey and +0-barreled.'' An example showing that the latter does not imply the former validates novelty, answers an old question, and completes a robust linear picture of ``Mackey weak barrelledness'' begun several decades ago. 1999 Academic Press Key Words: weak barrelledness; Mackey topology; splitting theorem. 1. INTRODUCTION Barrelled spaces have occupied an important place in the theory of locally convex spaces since its earliest days. This is probably due to the fact that they provide a vehicle for generalizing certain important properties of Banach spaces; for example, they form the class of domain spaces for a natural generalization of Banach's closed graph theorem and, in their dual characterization, they embody the conclusion of the BanachSteinhaus theorem for continuous linear functionals. Unlike Banach spaces, they are closed under the taking of inductive limits, countable-codimensional subspaces, products, etc. A barrel in a locally convex space is a closed * We are grateful for support and hospitality from EPSRC GRÂL67257 and Strathclyde University, and for helpful discussions with Professors J. -
A Generalisation of Mackey's Theorem and the Uniform Boundedness Principle
BULL. AUSTRAL. MATH. SOC. 46AO5 VOL. 40 (1989) [123-128] t A GENERALISATION OF MACKEY'S THEOREM AND THE UNIFORM BOUNDEDNESS PRINCIPLE CHARLES SWARTZ We construct a locally convex topology which is stronger than the Mackey topology but still has the same bounded sets as the Mackey topology. We use this topology to give a locally convex version of the Uniform Boundedness Principle which is valid without any completeness or barrelledness assumptions. A well known theorem of Mackey ([2, 20.11.7]) states that a subset of a locally convex topological vector space is bounded if and only if it is weakly bounded. In particular, this means that if two vector spaces X and X' are in duality, then the weak topology and the Mackey topology of X have the same bounded sets. In this note we construct a locally convex topology on X which is always stronger than the Mackey topology but which still has the same bounded sets as the Mackey (or the weak) topology. We give an example which shows that this topology which always has the same bounded sets as the Mackey topology can be strictly stronger than the Mackey topology and, therefore, can fail to be compatible with the duality between X and X'. As an application of this result, we give several versions of the Uniform Boundedness Principle for locally convex spaces which involve no completeness or barrelledness assumptions. For the remainder of this note, let X and X' be two vector spaces in duality with respect to the bilinear pairing <, > . The weak (Mackey, strong) topology on X will be denoted by cr(X, X')(T(X, X'), {3(X, X')). -
The Krein-Smulian Theorem
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 97. Number 2. June 1986 THE KREIN-SMULIAN THEOREM ROBERT WHITLEY In this note a brief and accessible proof of the Krein-Smulian theorem will be given. The lemma below is well known. The map J of the lemma is the canonical embedding of the Banach space X into its second dual X** defined by Jx(x*) = x*(x). Lemma. Let X be a separable Banach space and x** a linear functional in X**. Suppose that for each bounded sequence {x*} converging in the weak* topology of X* to x*, {x**(x*}} converges to x**(x*). Then x** belongs toJX. Proof. Let {*,.} be a dense subset of X. Suppose that d(x**, JX) = d > 0. By the Hahn-Banach theorem there is a norm one functional x*** satisfying x***(JX) = 0 and x***(x**) = d. Let W„ = {z*: \z*(x,)\ < 1 for / = 1,...,«}. By Gold- stine's theorem there is a functional x* in the unit ball of X* for which the values x**(x*), x*(xl),..., x*(x„) are as close as desired to the values x***(x**), x***(Jx1),..., x***(Jxn). Thus there is a functional x* belonging to [z*\ \\z*\\ < 1} Pi {z*: |jc**(z*)| > d/2) n Wn. The sequence {x*} converges to zero in the weak* topology, for given x and e > 0, there is an Xj with ||(x/e) - Xj\\ < 1, and thus |**(x)| < 2e for n > j; yet \x**(x*)\ > d/2 for all n. Q.E.D. Theorem (Krein-Smulian). The closed convex hull of a weakly compact subset of a Banach space is weakly compact. -
MATHEMATICAL TRIPOS Part III PAPER 106 FUNCTIONAL ANALYSIS
MATHEMATICAL TRIPOS Part III Monday, 30 May, 2016 9:00 am to 12:00 pm PAPER 106 FUNCTIONAL ANALYSIS Attempt no more than FOUR questions. There are FIVE questions in total. The questions carry equal weight. STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS Coversheet None Treasury Tag Script paper You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator. 2 1 Let A be a unital Banach algebra. Prove that the group G(A) of invertible elements of A is open and that the map x → x−1 : G(A) → G(A) is continuous. Let A be a Banach algebra and x ∈ A. Define the spectrum σA(x) of x in A both in the case that A is unital and in the case A is not unital. Let A be a Banach algebra and x ∈ A. Prove that σA(x) is a non-empty compact subset of C. [You may assume any version of the Hahn–Banach theorem without proof.] State and prove the Gelfand–Mazur theorem concerning complex unital normed division algebras. Let K be an arbitrary set. Let A be an algebra of complex-valued functions on K with pointwise operations and assume that · is a complete algebra norm on A. Prove carefully that supK|f| 6 f for all f ∈ A. 2 State and prove Mazur’s theorem on the weak-closure and norm-closure of a convex set in a Banach space. [Any version of the Hahn–Banach theorem can be assumed without proof provided it is clearly stated.] Let (xn) be a sequence in a Banach space X. -
A NOTE on NORM ATTAINING FUNCTIONALS 1. Notation And
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 7, July 1998, Pages 1989{1997 S 0002-9939(98)04739-X A NOTE ON NORM ATTAINING FUNCTIONALS M. JIMENEZ´ SEVILLA AND J. P. MORENO (Communicated by Palle E. T. Jorgensen) Abstract. We are concerned in this paper with the density of functionals which do not attain their norms in Banach spaces. Some earlier results given for separable spaces are extended to the nonseparable case. We obtain that a Banach space X is reflexive if and only if it satisfies any of the following properties: (i) X admits a norm with the Mazur Intersection Property k·k and the set NA of all norm attaining functionals of X∗ contains an open k·k set, (ii) the set NA1 of all norm one elements of NA contains a (relative) k·k k·k 1 weak* open set of the unit sphere, (iii) X∗ has C∗PCP and NA contains k·k a (relative) weak open set of the unit sphere, (iv) X is WCG, X∗ has CPCP and NA1 contains a (relative) weak open set of the unit sphere. Finally, if k·k X is separable, then X is reflexive if and only if NA1 contains a (relative) weak open set of the unit sphere. k·k 1. Notation and preliminaries Given a real Banach space (X, ) with dual (X∗, ∗), we denote by B its k·k k·k k·k closed unit ball and by S its unit sphere. A functional f X∗ attains its norm k·k ∈ if there exists x S such that f(x)= f ∗.Wedenoteby NA or simply by NA (if there is no∈ ambiguityk·k on the norm)k thek set of all norm attainingk·k functionals 1 1 of X∗ ,andby NA or NA the set of norm one elements of NA .