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Locally Solid Riesz Spaces with Applications to Economics / Charalambos D http://dx.doi.org/10.1090/surv/105 alambos D. Alipr Lie University \ Burkinshaw na University-Purdue EDITORIAL COMMITTEE Jerry L. Bona Michael P. Loss Peter S. Landweber, Chair Tudor Stefan Ratiu J. T. Stafford 2000 Mathematics Subject Classification. Primary 46A40, 46B40, 47B60, 47B65, 91B50; Secondary 28A33. Selected excerpts in this Second Edition are reprinted with the permissions of Cambridge University Press, the Canadian Mathematical Bulletin, Elsevier Science/Academic Press, and the Illinois Journal of Mathematics. For additional information and updates on this book, visit www.ams.org/bookpages/surv-105 Library of Congress Cataloging-in-Publication Data Aliprantis, Charalambos D. Locally solid Riesz spaces with applications to economics / Charalambos D. Aliprantis, Owen Burkinshaw.—2nd ed. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 105) Rev. ed. of: Locally solid Riesz spaces. 1978. Includes bibliographical references and index. ISBN 0-8218-3408-8 (alk. paper) 1. Riesz spaces. 2. Economics, Mathematical. I. Burkinshaw, Owen. II. Aliprantis, Char­ alambos D. III. Locally solid Riesz spaces. IV. Title. V. Mathematical surveys and mono­ graphs ; no. 105. QA322 .A39 2003 bib'.73—dc22 2003057948 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. © 2003 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 08 07 06 05 04 03 To the memory of my parents: Atou^ato KOLL YjUx\)povlot Charalambos Dionysios Aliprantis To my high school mathematics and science teachers, Elizabeth De Vary and Donald Roush. Great teachers make a difference in the lives of their students. Owen Burkinshaw Contents Preface of the 1st Edition vii Preface of the 2nd Edition ix List of Special Symbols xi Chapter 1. The Lattice Structure of Riesz spaces 1 1.1. Elementary Properties of Riesz spaces 1 1.2. Ideals, Bands, and Riesz Subspaces 10 1.3. Order Completeness and Projection Properties 18 1.4. The Freudenthal Spectral Theorem 25 1.5. The Main Inclusion Theorem 31 1.6. Order Bounded Operators 33 1.7. The Order Dual of a Riesz space 41 Chapter 2. Locally Solid Topologies 49 2.1. Linear Topologies on Vector Spaces 49 2.2. The Basic Properties of Locally Solid Topologies 55 2.3. Locally Convex-solid Topologies 59 2.4. Topological Completion of a Locally Solid Riesz Space 66 Chapter 3. Lebesgue Topologies 75 3.1. Examples and Properties of Lebesgue Topologies 75 3.2. Locally Convex-solid Lebesgue Topologies 80 3.3. Lebesgue Properties and Lp-Spaces 85 Chapter 4. Fatou Topologies 99 4.1. Basic Properties of the Fatou Topologies 99 4.2. The Structure of the Fatou Topologies 102 4.3. Topological Completeness and Fatou Topologies 108 4.4. Quotient Riesz Spaces and Fatou Properties 114 Chapter 5. Metrizability 119 5.1. Upper and Lower Elements 119 5.2. Frechet Topologies 125 5.3. The Pseudo Lebesgue Properties 129 5.4. The ^-Property 135 5.5. Comparing Locally Solid Topologies 136 Chapter 6. Weak Compactness in Riesz Spaces 143 6.1. Topologies on the Duals of a Riesz Space 143 6.2. Weak Compactness in the Order Duals 148 vi CONTENTS 6.3. Weak Sequential Convergence 158 6.4. Compact Solid Sets 162 6.5. Semireflexive Riesz Spaces 173 Chapter 7. Lateral Completeness 179 7.1. Laterally Complete Riesz Spaces 179 7.2. The Universal Completion 187 7.3. Lateral and Universal Completeness 196 7.4. Lateral Completeness and Local Solidness 201 7.5. Minimal Locally Solid Topologies 207 Chapter 8. Market Economies 215 8.1. Preferences and Utility Functions 215 8.2. Exchange Economies and Efficiency 217 8.3. Efficiency, Prices, and the Welfare Theorems 221 8.4. Properness 224 8.5. Properness and Efficiency 228 8.6. Equilibrium 231 8.7. Continuity Properties of Supporting Prices 233 8.8. The Utility Space of an Economy and Efficiency 236 8.9. Existence of Equilibria 240 8.10. The Core of an Economy 243 8.11. Replication 245 8.12. Edgeworth Equilibria 247 8.13. Core Equivalence 251 8.14. The Single Sector Growth Model 257 Chapter 9. Solutions to the Exercises 267 9.1. Chapter 1: The Lattice Structure of Riesz Spaces 267 9.2. Chapter 2: Locally Solid Topologies 279 9.3. Chapter 3: Lebesgue Topologies 284 9.4. Chapter 4: Fatou Topologies 289 9.5. Chapter 5: Metrizability 295 9.6. Chapter 6: Weak Compactness in Riesz Spaces 301 9.7. Chapter 7: Lateral Completeness 307 9.8. Chapter 8: Market Economies 317 Bibliography 331 Index 337 Preface of the 1st Edition In 1928, at the International Mathematical Congress in Bologna, Italy, F. Riesz in a short address [115] triggered the investigation of what is today called the theory of Riesz spaces. Soon after, in the mid-thirties, H. Freudenthal [60] and L. V. Kantorovich [69, 70] independently set up the axiomatic foundation and derived a number of properties dealing with the lattice structure of Riesz spaces. From then on the growth of the subject was rapid. In the forties and early fifties the Japanese school led by H. Nakano, T. Ogasawara, and K. Yosida, and the Russian school, led by L. V. Kantorovich, A. I. Judin, and B. Z. Vulikh, made fundamental contributions. At the same time a number of books started to appear on the field. Most of the early work on Riesz spaces, as well as most of the books, was devoted to the so-called "algebraic part" of the theory and little attention was given to the "analytical part." The bulk of the existing work on the analytical part is in the area of normed Riesz spaces, and in particular in the theory of Banach lattices; we note the work of W. A. J. Luxemburg and A. C. Zaanen appearing in a series of articles [89] and the recent book of H. H. Schaefer [123]. The general theory of topological Riesz spaces seems somehow to have been neglected. The recent book by D. H. Fremlin [58] is partially devoted to this subject; and there are, of course, a number of books on linear ordered spaces (see [65, 108, 127, 130]). But there have been no books completely devoted to topological Riesz spaces. The purpose of this book is to fill this gap and present a unified approach to the theory of locally solid Riesz spaces. It has been our desire to emphasize the relationship between the order and topological structures. The present work is written for graduate students and for persons involved in research in functional analysis. The book is self-contained and should be accessible to any student with a standard course in functional analysis. The material has been arranged into seven chapters. As a supplement, at the end of each chapter, we have included a number of exercises of varying degrees of difficulty. We have also listed some open problems to give insight into possible areas of research. Throughout the book the basic concepts are illustrated by numerous examples and counterexamples, and special notes trace the history of the subject. However, it is possible that credit was not given to all contributors to this field. In Chapter 1 a summary of the basic theory of the lattice structure of Riesz spaces is given, with special emphasis on results that are not easily accessible and to proofs that are not well known. The fundamental properties of locally solid topologies are presented in Chapter 2, while Chapter 3 studies the interaction between order convergence and topological convergence (Lebesgue properties) with applications to the representation theorems of abstract Lp-spaces. vii st Vlll PREFACE OF THE 1 EDITION Chapter 4 deals with Fatou topologies and shows the remarkable relationship between order completeness and topological completeness, while Chapter 5 inves­ tigates the important properties of metrizable locally solid Riesz spaces. In Chapter 6 locally solid topologies that are also locally convex are considered. Weakly compact sets are studied via the concept of order-equicontinuity. The interplay between the order structure and the topological concept of compactness is demonstrated extensively, and characterizations of semireflexive Riesz spaces are presented. Finally, Chapter 7 discusses both the remarkable lattice and topological properties of laterally complete Riesz spaces. This book has been influenced by the discussions the first author had with W. A. J. Luxemburg and the second with S. Kaplan, to whom we express our appreciation for introducing us to the subject. Finally, we would like to thank our wives, Bernadette and Betty, for their help and understanding during the preparation of this book. Indianapolis C. D. ALIPRANTIS June 1977 O. BURKINSHAW Preface of the 2nd Edition This monograph is the 2nd edition of our book Locally Solid Riesz Spaces that was published by Academic Press in 1978.
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