The Three Space Problem for Locally Bounded F-Spaces Compositio Mathematica, Tome 37, No 3 (1978), P
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
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. -
209 a Note on Closed Graph Theorems
Acta Math. Univ. Comenianae 209 Vol. LXXV, 2(2006), pp. 209–218 A NOTE ON CLOSED GRAPH THEOREMS F. GACH Abstract. We give a common generalisation of the closed graph theorems of De Wilde and of Popa. 1. Introduction In the theory of locally convex spaces, M. De Wilde’s notion of webs is the abstrac- tion of all that is essential in order to prove very general closed graph theorems. Here we concentrate on his theorem in [1] for linear mappings with bornologically closed graph that have an ultrabornological space as domain and a webbed lo- cally convex space as codomain. Henceforth, we shall refer to this theorem as ‘De Wilde’s closed graph theorem’. As far as the category of bounded linear mappings between separated convex bornological spaces is concerned, there exists a corresponding bornological notion of so-called nets that enabled N. Popa in [5] to prove a bornological version of the closed graph theorem which we name ‘Popa’s closed graph theorem’. Although M. De Wilde’s theorem in the locally convex and N. Popa’s theorem in the convex bornological setting are conceptually similar (see also [3]), they do not directly relate to each other. In this manuscript I present a bornological closed graph theorem that generalises the one of N. Popa and even implies the one of M. De Wilde. First of all, I give a suitable definition of bornological webs on separated convex bornological spaces with excellent stability properties and prove the aforementioned general bornological closed graph theorem. It turns out that there is a connection between topological webs in the sense of M. -
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. -
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. -
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. -
Some Simple Applications of the Closed Graph Theorem
SHORTER NOTES The purpose of this department is to publish very short papers of an unusually elegant and polished character, for which there is normally no other outlet. SOME SIMPLE APPLICATIONS OF THE CLOSED GRAPH THEOREM JESÚS GIL DE LAMADRID1 Our discussion is based on the following two simple lemmas. Our field of scalars can be either the reals or the complex numbers. Lemma 1. Let Ebea Banach space under a norm || || and Fa normed space under a norm || ||i. Suppose that || H2is a second norm of F with respect to which F is complete and such that || ||2^^|| \\ifor some k>0. Assume finally that T: E—+Fis a linear transformation which is bounded with respect toll II and II IK. Then T is bounded with respect to\\ \\ and Proof. The product topology oî EX F with respect to || || and ||i is weaker (coarser) than the product topology with respect to || and || 112.Since T is bounded with respect to j| || and || ||i, its graph G is (|| | , II ||i)-closed. Hence G is also (|| ||, || ||2)-closed. Therefore T is ( | [[, || ||2)-bounded by the closed graph theorem. Lemma 2. Assume that E is a Banach space under a norm || ||i and that Fis a vector subspace2 of E, which is a Banach space under a second norm || ||2=è&|| ||i, for some ft>0. Suppose that T: E—>E is a bounded operator with respect to \\ \\i and \\ ||i smcA that T(F)EF. Then the re- striction of T to F is bounded with respect to || H2and || ||2. Proof. -
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. -
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 . -
Metrizable Barrelled Spaces, by J . C. Ferrando, M. López Pellicer, And
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 34, Number 1, January 1997 S 0273-0979(97)00706-4 Metrizable barrelled spaces,byJ.C.Ferrando,M.L´opez Pellicer, and L. M. S´anchez Ruiz, Pitman Res. Notes in Math. Ser., vol. 332, Longman, 1995, 238 pp., $30.00, ISBN 0-582-28703-0 The book targeted by this review is the next monograph written by mathemati- cians from Valencia which deals with barrelledness in (metrizable) locally convex spaces (lcs) over the real or complex numbers and applications in measure theory. This carefully written and well-documented book complements excellent mono- graphs by Valdivia [21] and by P´erez Carreras, Bonet [12] in this area. Although for the reader some knowledge of locally convex spaces is requisite, the book under re- view is self-contained with all proofs included and may be of interest to researchers and graduate students interested in locally convex spaces, normed spaces and mea- sure theory. Most chapters contain some introductory facts (with proofs), which for the reader greatly reduces the need for original sources. Also a stimulating Notes and Remarks section ends each chapter of the book. The classical Baire category theorem says that if X is either a complete metric space or a locally compact Hausdorff space, then the intersection of countably many dense, open subsets of X is a dense subset of X. This theorem is a principal one in analysis, with applications to well-known theorems such as the Closed Graph Theorem, the Open Mapping Theorem and the Uniform Boundedness Theorem. -
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.