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Subspaces and Quotients of Topological and Ordered Vector Spaces Zoran Kadelburg Stojan Radenovi´c SUBSPACES AND QUOTIENTS OF TOPOLOGICAL AND ORDERED VECTOR SPACES Novi Sad, 1997. CONTENTS INTRODUCTION::::::::::::::::::::::::::::::::::::::::::::::::::::: 1 I: TOPOLOGICAL VECTOR SPACES::::::::::::::::::::::::::::::: 3 1.1. Some properties of subsets of vector spaces ::::::::::::::::::::::: 3 1.2. Topological vector spaces::::::::::::::::::::::::::::::::::::::::: 6 1.3. Locally convex spaces :::::::::::::::::::::::::::::::::::::::::::: 12 1.4. Inductive and projective topologies ::::::::::::::::::::::::::::::: 15 1.5. Topologies of uniform convergence. The Banach-Steinhaus theorem 21 1.6. Duality theory ::::::::::::::::::::::::::::::::::::::::::::::::::: 28 II: SUBSPACES AND QUOTIENTS OF TOPOLOGICAL VECTOR SPACES ::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 39 2.1. Subspaces of lcs’s belonging to the basic classes ::::::::::::::::::: 39 2.2. Subspaces of lcs’s from some other classes :::::::::::::::::::::::: 47 2.3. Subspaces of topological vector spaces :::::::::::::::::::::::::::: 56 2.4. Three-space-problem for topological vector spaces::::::::::::::::: 60 2.5. Three-space-problem in Fr´echet spaces:::::::::::::::::::::::::::: 65 III: ORDERED TOPOLOGICAL VECTOR SPACES :::::::::::::::: 72 3.1. Basics of the theory of Riesz spaces::::::::::::::::::::::::::::::: 72 3.2. Topological vector Riesz spaces ::::::::::::::::::::::::::::::::::: 79 3.3. The basic classes of locally convex Riesz spaces ::::::::::::::::::: 82 3.4. l-ideals of topological vector Riesz spaces ::::::::::::::::::::::::: 90 3.5. 3SP in the class of locally convex Riesz spaces :::::::::::::::::::: 92 IV: SUPPLEMENT: ENLARGEMENTS OF LOCALLY CONVEX TOPOLOGIES :::::::::::::::::::::::::::::::::::::::::::::::::: 100 4.1. Finite-dimensional enlargements :::::::::::::::::::::::::::::::::: 100 4.2. Countable enlargements of barrelled topologies ::::::::::::::::::: 103 4.3. Satisfactory subspaces of barrelled spaces ::::::::::::::::::::::::: 106 4.4. Open problems::::::::::::::::::::::::::::::::::::::::::::::::::: 112 REFERENCES:::::::::::::::::::::::::::::::::::::::::::::::::::::::: 113 SUBJECT INDEX :::::::::::::::::::::::::::::::::::::::::::::::::::: 120 INTRODUCTION First results concerning not necessarily normed topological vector spaces date back to middle thirties of this century. But it was not until late forties and early fifties that the theory of such spaces was formed as a special branch in functional analysis. This was the time when it became clear that a lot of applications, in particular the theory of distributions, need more abstract approach than the classical Banach one. Within the framework of the new categories of topological vector, in particular locally convex, spaces and, a bit later, of ordered topological vector spaces, a lot of special classes of spaces were introduced. The reason was mainly that the three fundamental principles of linear analysis, in particular the uniform boundedness principle and the closed graph and open mapping theorems, needed special types of spaces to serve as domains and/or codomains for respective mappings. Such classes of spaces, e.g. barrelled, quasibarrelled, bornological and many others, proved themselves to be important in connection with other problems, too, such as problems in duality theory, reflexivity, tensor products and so on. One of the important questions concerning these classes of spaces was their stability in connection with subspaces and quotients. While most of them (but not all) are stable with respect to quotients by closed subspaces, few are stable with respect to arbitrary, even closed subspaces. So it was natural to search for various sufficient conditions which would imply stabilities of such kind. First results of that type were obtained in early fifties, but investigations have continued and there are a lot of interesting recent results. Some questions even in pure Banach setting are still open, e.g. “is there a Banach space of infinite dimension such that all of its dense subspaces are barrelled” or “does every infinite-dimensional Banach space have a separable quotient”? This book is divided into four chapters. The first one contains a very brief introduction to the theory of topological vector spaces. The intention was to give only the fundamental ideas of the theory, particularly those which will be used in the rest of the book. Some important theorems are stated without proofs, and the reader is referred to the classical books covering these problems. But we hope that this will be enough to get prepared for reading the remaining three chapters. Especially, at least formally speaking, the prerequisites for the whole book are just the basics of linear algebra and general topology, although, of course, the acquaintance with classical functional analysis is desirable. 2 Introduction Chapter II treats thoroughly the problems of subspaces and quotients for various classes of topological vector, in particular locally convex, spaces. Also, a kind of a converse question, the so-called three-space-problem, is investigated. Some of the results of the authors are included in this and the following chapter. In many applications of functional analysis, besides linear and topological structures, also the structure of order is often important. So a special theory of ordered vector spaces, and in particular ordered topological vector spaces, has been developed. Our third chapter treats the basics of this theory and in particular those concepts which are close to the ones described in the first part of the book. It is shown that the richer structure enables more positive results than in the category of vector spaces without order. There are a lot of problems and streams of investigation which are actual nowadays in the theory of topological vector spaces. We chose some of them and included as a supplement in the chapter IV. We hope that this will help to get an idea in what directions the theory may be developed and, maybe, some of the readers will try to do something on their own in this field. A brief list of open questions which is given at the end might help. Although the list of references is rather long, it is very far from being complete, even when the special topics of this book are concerned. We mentioned just those titles which we referred to directly and, with a few exceptions, we didn’t mention articles which are already included in existing monographs. We are very grateful to the Institute of Mathematics of the University of Novi Sad for publishing this monograph. Beograd, October 1996 Authors I. TOPOLOGICAL VECTOR SPACES 1.1. SOME PROPERTIES OF SUBSETS OF VECTOR SPACES All vector space in this book are supposed to be defined over the fixed field K, which is either the field R of real or the field C of complex numbers. Only in the third chapter just real scalars are used. In this paragraph we shall describe some properties of subsets of vector spaces which are important in the theory of topological and ordered vector spaces. Let E be a vector space (over the field K) and let A be its subset. We say that A is: 1± circled if ¸A ½ A for each ¸ 2 K for which j¸j 6 1; 2± convex if ¸x + (1 ¡ ¸)y 2 A for each x; y 2 A and each ¸ 2 K, 0 6 ¸ 6 1; 3± absolutely convex if ¸x + ¹y 2 A for each x; y 2 A and each ¸; ¹ 2 K for which j¸j + j¹j 6 1. It is clear that the intersection of an arbitrary family of (absolutely) convex subsets is again an (absolutely) convex set. The (absolutely) convex cover of the set A ½ E is the intersection of all (absolutely) convex sets containig A; we shall denote it by co A (resp. ΓA). If (Ai), Si 2 I is a familyS of subsets of E , we shall write coi Ai (resp. ΓiAi ) instead of co( i Ai) (resp. Γ( i Ai)). Pn Proposition 1.1. co A is the set of all elements of the form ¸ixi for 1 Pn Pn ¸i > 0, ¸i = 1, xi 2 A. ΓA is the set of all elements of the form ¸ixi for 1 1 Pn ¸i 2 K, j¸ij 6 1, xi 2 A. 1 Proof. We shall prove the assertion for the convex cover; for the absolutely convex cover the proof is similar. It can be checked directly that the elements of the given form create a set which is convex and contains A. Let us show that, conversely, these elements belong to each convex set which contains A. nP¡1 Suppose that the last assertion is proved for each sum of the form ¸ixi . 1 Pn nP¡1 Let ¹ixi be a combination of the given form and let ¹i = ¹ > 0 (for ¹ = 0 1 1 4 I. Topological vector spaces n¡1 P ¹i the assertion is trivial). By the inductive hypothesis, y = xi 2 co A, and so 1 ¹ Pn ¹y + (1 ¡ ¹)xn = ¹ixi 2 co A since 0 < ¹ 6 1. 1 By the circled cover of a set A ½ E we understand the intersection of all circled subsets of E which contain A. Proposition 1.2. The absolutely convex cover of a subset is the convex cover of its circled cover. A set is absolutely convex if and only if it is convex and circled. The circled cover of the convex cover of a set need not be convex. Proof. Since, obviously, each absolutely convex set is circled, it is enough to prove that the convex cover of a circled set A is absolutely convex. According to Proposition 1.1, we have co A ½ ΓA. Conversely, let ¸i 6= 0 from K be given, such Pn Pn Pn that j¸ij 6 1 and let xi 2 A. Denote ¹i = (¸i=j¸ij) j¸ij and ºi = j¸ij= j¸ij. 1 1 1 Pn Pn Then each x = j¸ijxi 2 ΓA can be written in the form x = ºi¹ixi , where 1 1 Pn ºi = 1, ºi > 0 and j¹ij 6 1, ¹ixi 2 A, so that x 2 co A. Thus ΓA ½ co A. 1 Let A = f(0; 0); (1; 0); (1; 1)g, treated as a subset either of R2 or C2 . It is easy to check that the circled cover of the set co A is not convex. ± Proposition 1.3. 1 If A1 , ::: , An are (absolutely) convex sets and ¸1 , ::: , ¸n 2 K, then ¸1A1 + ¢ ¢ ¢ + ¸nAn is an (absolutely) convex set.
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