Eberlein-Šmulian Theorem and Some of Its Applications
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On Quasi Norm Attaining Operators Between Banach Spaces
ON QUASI NORM ATTAINING OPERATORS BETWEEN BANACH SPACES GEUNSU CHOI, YUN SUNG CHOI, MINGU JUNG, AND MIGUEL MART´IN Abstract. We provide a characterization of the Radon-Nikod´ymproperty in terms of the denseness of bounded linear operators which attain their norm in a weak sense, which complement the one given by Bourgain and Huff in the 1970's. To this end, we introduce the following notion: an operator T : X ÝÑ Y between the Banach spaces X and Y is quasi norm attaining if there is a sequence pxnq of norm one elements in X such that pT xnq converges to some u P Y with }u}“}T }. Norm attaining operators in the usual (or strong) sense (i.e. operators for which there is a point in the unit ball where the norm of its image equals the norm of the operator) and also compact operators satisfy this definition. We prove that strong Radon-Nikod´ymoperators can be approximated by quasi norm attaining operators, a result which does not hold for norm attaining operators in the strong sense. This shows that this new notion of quasi norm attainment allows to characterize the Radon-Nikod´ymproperty in terms of denseness of quasi norm attaining operators for both domain and range spaces, completing thus a characterization by Bourgain and Huff in terms of norm attaining operators which is only valid for domain spaces and it is actually false for range spaces (due to a celebrated example by Gowers of 1990). A number of other related results are also included in the paper: we give some positive results on the denseness of norm attaining Lipschitz maps, norm attaining multilinear maps and norm attaining polynomials, characterize both finite dimensionality and reflexivity in terms of quasi norm attaining operators, discuss conditions to obtain that quasi norm attaining operators are actually norm attaining, study the relationship with the norm attainment of the adjoint operator and, finally, present some stability results. -
Uniform Boundedness Principle for Unbounded Operators
UNIFORM BOUNDEDNESS PRINCIPLE FOR UNBOUNDED OPERATORS C. GANESA MOORTHY and CT. RAMASAMY Abstract. A uniform boundedness principle for unbounded operators is derived. A particular case is: Suppose fTigi2I is a family of linear mappings of a Banach space X into a normed space Y such that fTix : i 2 Ig is bounded for each x 2 X; then there exists a dense subset A of the open unit ball in X such that fTix : i 2 I; x 2 Ag is bounded. A closed graph theorem and a bounded inverse theorem are obtained for families of linear mappings as consequences of this principle. Some applications of this principle are also obtained. 1. Introduction There are many forms for uniform boundedness principle. There is no known evidence for this principle for unbounded operators which generalizes classical uniform boundedness principle for bounded operators. The second section presents a uniform boundedness principle for unbounded operators. An application to derive Hellinger-Toeplitz theorem is also obtained in this section. A JJ J I II closed graph theorem and a bounded inverse theorem are obtained for families of linear mappings in the third section as consequences of this principle. Go back Let us assume the following: Every vector space X is over R or C. An α-seminorm (0 < α ≤ 1) is a mapping p: X ! [0; 1) such that p(x + y) ≤ p(x) + p(y), p(ax) ≤ jajαp(x) for all x; y 2 X Full Screen Close Received November 7, 2013. 2010 Mathematics Subject Classification. Primary 46A32, 47L60. Key words and phrases. -
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. -
Functional Analysis Lecture Notes Chapter 3. Banach
FUNCTIONAL ANALYSIS LECTURE NOTES CHAPTER 3. BANACH SPACES CHRISTOPHER HEIL 1. Elementary Properties and Examples Notation 1.1. Throughout, F will denote either the real line R or the complex plane C. All vector spaces are assumed to be over the field F. Definition 1.2. Let X be a vector space over the field F. Then a semi-norm on X is a function k · k: X ! R such that (a) kxk ≥ 0 for all x 2 X, (b) kαxk = jαj kxk for all x 2 X and α 2 F, (c) Triangle Inequality: kx + yk ≤ kxk + kyk for all x, y 2 X. A norm on X is a semi-norm which also satisfies: (d) kxk = 0 =) x = 0. A vector space X together with a norm k · k is called a normed linear space, a normed vector space, or simply a normed space. Definition 1.3. Let I be a finite or countable index set (for example, I = f1; : : : ; Ng if finite, or I = N or Z if infinite). Let w : I ! [0; 1). Given a sequence of scalars x = (xi)i2I , set 1=p jx jp w(i)p ; 0 < p < 1; 8 i kxkp;w = > Xi2I <> sup jxij w(i); p = 1; i2I > where these quantities could be infinite.:> Then we set p `w(I) = x = (xi)i2I : kxkp < 1 : n o p p p We call `w(I) a weighted ` space, and often denote it just by `w (especially if I = N). If p p w(i) = 1 for all i, then we simply call this space ` (I) or ` and write k · kp instead of k · kp;w. -
The Nonstandard Theory of Topological Vector Spaces
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 172, October 1972 THE NONSTANDARDTHEORY OF TOPOLOGICAL VECTOR SPACES BY C. WARD HENSON AND L. C. MOORE, JR. ABSTRACT. In this paper the nonstandard theory of topological vector spaces is developed, with three main objectives: (1) creation of the basic nonstandard concepts and tools; (2) use of these tools to give nonstandard treatments of some major standard theorems ; (3) construction of the nonstandard hull of an arbitrary topological vector space, and the beginning of the study of the class of spaces which tesults. Introduction. Let Ml be a set theoretical structure and let *JR be an enlarge- ment of M. Let (E, 0) be a topological vector space in M. §§1 and 2 of this paper are devoted to the elementary nonstandard theory of (F, 0). In particular, in §1 the concept of 0-finiteness for elements of *E is introduced and the nonstandard hull of (E, 0) (relative to *3R) is defined. §2 introduces the concept of 0-bounded- ness for elements of *E. In §5 the elementary nonstandard theory of locally convex spaces is developed by investigating the mapping in *JK which corresponds to a given pairing. In §§6 and 7 we make use of this theory by providing nonstandard treatments of two aspects of the existing standard theory. In §6, Luxemburg's characterization of the pre-nearstandard elements of *E for a normed space (E, p) is extended to Hausdorff locally convex spaces (E, 8). This characterization is used to prove the theorem of Grothendieck which gives a criterion for the completeness of a Hausdorff locally convex space. -
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. -
Arxiv:2003.03538V1 [Math.FA] 7 Mar 2020 Ooois Otniy Discontinuity
CONTINUITY AND DISCONTINUITY OF SEMINORMS ON INFINITE-DIMENSIONAL VECTOR SPACES. II JACEK CHMIELINSKI´ Department of Mathematics, Pedagogical University of Krak´ow Krak´ow, Poland E-mail: [email protected] MOSHE GOLDBERG1 Department of Mathematics, Technion – Israel Institute of Technology Haifa, Israel E-mail: [email protected] Abstract. In this paper we extend our findings in [3] and answer further questions regard- ing continuity and discontinuity of seminorms on infinite-dimensional vector spaces. Throughout this paper let X be a vector space over a field F, either R or C. As usual, a real-valued function N : X → R is a norm on X if for all x, y ∈ X and α ∈ F, N(x) > 0, x =06 , N(αx)= |α|N(x), N(x + y) ≤ N(x)+ N(y). Furthermore, a real-valued function S : X → R is called a seminorm if for all x, y ∈ X and α ∈ F, arXiv:2003.03538v1 [math.FA] 7 Mar 2020 S(x) ≥ 0, S(αx)= |α|S(x), S(x + y) ≤ S(x)+ S(y); hence, a norm is a positive-definite seminorm. Using standard terminology, we say that a seminorm S is proper if S does not vanish identically and S(x) = 0 for some x =6 0 or, in other words, if ker S := {x ∈ X : S(x)=0}, is a nontrivial proper subspace of X. 2010 Mathematics Subject Classification. 15A03, 47A30, 54A10, 54C05. Key words and phrases. infinite-dimensional vector spaces, Banach spaces, seminorms, norms, norm- topologies, continuity, discontinuity. 1Corresponding author. 1 2 JACEK CHMIELINSKI´ AND MOSHE GOLDBERG Lastly, just as for norms, we say that seminorms S1 and S2 are equivalent on X, if there exist positive constants β ≤ γ such that for all x ∈ X, βS1(x) ≤ S2(x) ≤ γS1(x). -
1 the Principal of Uniform Boundedness
THE PRINCIPLE OF UNIFORM BOUNDEDNESS 1 The Principal of Uniform Boundedness Many of the most important theorems in analysis assert that pointwise hy- potheses imply uniform conclusions. Perhaps the simplest example is the theorem that a continuous function on a compact set is uniformly contin- uous. The main theorem in this section concerns a family of bounded lin- ear operators, and asserts that the family is uniformly bounded (and hence equicontinuous) if it is pointwise bounded. We begin by defining these terms precisely. Definition 1.1 Let A be a family of linear operators from a normed space X to a normed space Y . We say that A is pointwise bounded if supA2AfkAxkg < 1 for every x 2 X. We say A is uniformly bounded if supA2AfkAkg < 1. It is possible for a single linear operator to be pointwise bounded without being bounded (Exercise), so the hypothesis in the next theorem that each individual operator is bounded is essential. Theorem 1.2 (The Principle of Uniform Boundedness) Let A ⊆ L(X; Y ) be a family of bounded linear operators from a Banach space X to a normed space Y . Then A is uniformly bounded if and only if it is pointwise bounded. Proof We will assume that A is pointwise bounded but not uniformly bounded, and obtain a contradiction. For each x 2 X, define M(x) := supA2AfkAxkg; our assumption is that M(x) < 1 for every x. Observe that if A is not uniformly bounded, then for any pair of positive numbers and C there must exist some A 2 A with kAk > C/, and hence some x 2 X with kxk = but kAxk > C. -
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]. -
The Banach-Alaoglu Theorem for Topological Vector Spaces
The Banach-Alaoglu theorem for topological vector spaces Christiaan van den Brink a thesis submitted to the Department of Mathematics at Utrecht University in partial fulfillment of the requirements for the degree of Bachelor in Mathematics Supervisor: Fabian Ziltener date of submission 06-06-2019 Abstract In this thesis we generalize the Banach-Alaoglu theorem to topological vector spaces. the theorem then states that the polar, which lies in the dual space, of a neighbourhood around zero is weak* compact. We give motivation for the non-triviality of this theorem in this more general case. Later on, we show that the polar is sequentially compact if the space is separable. If our space is normed, then we show that the polar of the unit ball is the closed unit ball in the dual space. Finally, we introduce the notion of nets and we use these to prove the main theorem. i ii Acknowledgments A huge thanks goes out to my supervisor Fabian Ziltener for guiding me through the process of writing a bachelor thesis. I would also like to thank my girlfriend, family and my pet who have supported me all the way. iii iv Contents 1 Introduction 1 1.1 Motivation and main result . .1 1.2 Remarks and related works . .2 1.3 Organization of this thesis . .2 2 Introduction to Topological vector spaces 4 2.1 Topological vector spaces . .4 2.1.1 Definition of topological vector space . .4 2.1.2 The topology of a TVS . .6 2.2 Dual spaces . .9 2.2.1 Continuous functionals . -
Interpolation of Multilinear Operators Acting Between Quasi-Banach Spaces
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 142, Number 7, July 2014, Pages 2507–2516 S 0002-9939(2014)12083-1 Article electronically published on April 8, 2014 INTERPOLATION OF MULTILINEAR OPERATORS ACTING BETWEEN QUASI-BANACH SPACES L. GRAFAKOS, M. MASTYLO, AND R. SZWEDEK (Communicated by Alexander Iosevich) Abstract. We show that interpolation of multilinear operators can be lifted to multilinear operators from spaces generated by the minimal methods to spaces generated by the maximal methods of interpolation defined on a class of couples of compatible p -Banach spaces. We also prove the multilinear interpolation theorem for operators on Calder´on-Lozanovskii spaces between Lp-spaces with 0 <p≤ 1. As an application we obtain interpolation theorems for multilinear operators on quasi-Banach Orlicz spaces. 1. Introduction In the study of the many problems which appear in various areas of analysis it is essential to know whether important operators are bounded between certain quasi-Banach spaces. Motivated in particular by applications in harmonic analysis, we are interested in proving new abstract multilinear interpolation theorems for multilinear operators between quasi-Banach spaces. Based on ideas from the theory of operators between Banach spaces, we use the universal method of interpolation defined on proper classes of quasi-Banach spaces. It should be pointed out that in general the interpolation methods used in the case of Banach spaces do not apply in the setting of quasi-Banach spaces. The main reason is that the topological dual spaces of quasi-Banach spaces could be trivial and the same may be true for spaces of continuous linear operators between spaces from a wide class of quasi-Banach spaces. -
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.