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View This Volume's Front and Back Matter http://dx.doi.org/10.1090/gsm/084 Cone s an d Dualit y This page intentionally left blank Cone s an d Dualit y Charalambo s D . Alipranti s RabeeTourk y Graduate Studies in Mathematics Volum e 84 •& Ip^Sn l America n Mathematica l Societ y *0||jjO ? provjcjence i Rhod e Islan d Editorial Board David Cox (Chair) Walter Craig N. V. Ivanov Steven G. Krantz 2000 Mathematics Subject Classification. Primary 46A40, 46B40, 47B60, 47B65; Secondary 06F30, 28A33, 91B28, 91B99. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-84 Library of Congress Cataloging-in-Publication Data Aliprantis, Charalambos D. Cones and duality / Charalambos D. Aliprantis, Rabee Tourky. p. cm. — (Graduate studies in mathematics, ISSN 1065-7339 ; v. 84) Includes bibliographical references and index. ISBN 978-0-8218-4146-4 (alk. paper) 1. Cones (Operator theory). 2. Linear topological spaces, Ordered. I. Tourky, Rabee, 1966- II. Title. QA329 .A45 2007 515'.724—dc22 2007060758 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]. © 2007 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 12 11 10 09 08 07 To the great Russian mathematician and economist Leonid Vitaliyevich Kantorovich (1912-1986), the 1975 Nobel Prize co-recipient in economics, ... whose brilliant ideas have shaped the field of ordered vector spaces and are present throughout this book. This page intentionally left blank Contents Preface The "isomorphism" notion Chapter 1. Cones §1.1. Wedges and cones §1.2. Archimedean cones §1.3. Lattice cones §1.4. Positive and order bounded operators §1.5. Positive linear functionals §1.6. Faces and extremal vectors of cones §1.7. Cone bases §1.8. Decomposability in ordered vector spaces §1.9. The Riesz-Kantorovich formulas Chapter 2. Cones in topological vector spaces §2.1. Ordered topological vector spaces §2.2. Wedge duality §2.3. Normal cones §2.4. Positivity and continuity §2.5. Ordered Banach spaces §2.6. Ice cream cones in normed spaces §2.7. Ideals in ordered vector spaces §2.8. The order topology generated by a cone Vlll Contents Chapter 3. Yudin and pull-back cones 117 §3.1. Closed cones in finite dimensional vector spaces 118 §3.2. Directional wedges and Yudin cones 122 §3.3. Polyhedral wedges and cones 131 §3.4. The geometrical structure of polyhedral cones 137 §3.5. Linear inequalities and alternatives 148 §3.6. Pull-back cones of operators 152 Chapter 4. Krein operators 159 §4.1. The concept of a Krein operator 160 §4.2. Eigenvalues of Krein operators 163 §4.3. Fixed points and eigenvectors 167 Chapter 5. K-lattices 173 §5.1. The notion and properties of K-lattices 174 §5.2. The Riesz–Kantorovich transform 183 §5.3. The order extension of £b(L, N) 190 Chapter 6. The order extension of V 197 §6.1. The extension of V 199 §6.2. Generalized Riesz-Kantorovich functionals 204 §6.3. When is the Riesz-Kantorovich functional additive? 210 Chapter 7. Piecewise affine functions 221 §7.1. One-dimensional piecewise affine functions 221 §7.2. Multivariate piecewise affine functions 227 Chapter 8. Appendix: linear topologies 243 §8.1. Linear topologies on vector spaces 244 §8.2. Duality theory 247 §8.3. 6-topologies 249 §8.4. The separation of convex sets 251 §8.5. Normed and Banach spaces 252 §8.6. Finite dimensional topological vector spaces 256 §8.7. The open mapping and the closed graph theorems 257 §8.8. The bounded weak* topology 259 Bibliography 265 Index 271 Preface Ordered vector spaces made their debut at the beginning of the twentieth century. They were developed in parallel (but from a different perspec• tive) with functional analysis and operator theory. Before the 1950s ordered vector spaces appeared in the literature in a fragmented way. Their sys• tematic study began in various schools around the world after the 1950s. We mention the Russian school (headed by L. V. Kantorovich and the Krein brothers), the Japanese school (headed by H. Nakano), the Ger• man school (headed by H. H. Schaefer), and the Dutch school (headed by A. C. Zaanen). At the same time several monographs dealing exclu• sively with ordered vector spaces appeared in the literature; see for in• stance [55, 56, 71, 75, 89, 91]. The special class of ordered vector spaces known as Riesz spaces or vector lattices has been studied more extensively; see the monographs [14, 15, 66, 68, 86, 88, 93]. The theory of ordered vector spaces plays a prominent role in functional analysis. It also contributes to a wide variety of applications and is an indispensable tool for studying a variety of problems in engineering and economics; see for instance [29, 31, 35, 36, 38, 42, 47, 49, 54, 64, 65, 76]. The introduction of Riesz spaces and more broadly ordered vector spaces to economic theory has proved tremendously successful and has allowed researchers to answer difficult questions in general price equilibrium theory, economies with differential information, the theory of perfect competition, and incomplete assets economies. The goal of this monograph is to present the theory of ordered vector spaces from a contemporary perspective that has been influenced by the study of ordered vector spaces in economics as well as other recent appli• cations. We try to imbue the narrative with geometric intuition, which is IX X Preface in keeping with a long tradition in mathematical economics. We also ap• proach the subject with our own personal presentiment that the special class of Riesz spaces is somehow "perfect" and thus loosely conceive of general ordered vector spaces as "deviations" from this "perfection." The book also contains material that has not been published in a monograph form before. The study of this material was initially motivated by various problems in economics and econometrics. The material is spread out in eight chapters. Chapter 8 is an Appendix and contains some basic notions of functional analysis. Special attention is paid to the properties of linear topologies and the separation of convex sets. The results in this chapter (some of which are presented with proofs) are used throughout the monograph without specific mention. Chapter 1 presents the fundamental properties of wedges and cones. Here we discuss Archimedean cones, lattice cones, extremal vectors of cones, bases of cones, positive linear functionals and the important decomposabil- ity property of cones known as the Riesz decomposition property. Chapter 2 introduces cones in topological vector spaces. This chapter illustrates the variety of remarkable results that can be obtained when some link between the order and the topology is imposed. The most important interrelationship between a cone and a linear topology is known as normality. We discuss nor• mal cones in detail and obtain several characterizations. In normed spaces, the normality of the cone amounts to the norm boundedness of the order intervals generated by the cone. In Chapter 2 we also introduce ideals and present some of their useful order and topological properties. Chapter 3 studies in detail cones in finite dimensional vector spaces. The results here are much sharper. For instance, as we shall see, every closed cone of a finite dimensional vector space is normal. The reader will find in this chapter a study (together with a geometrical description) of the polyhedral cones as well as a discussion of the properties of linear inequalities—including a proof of "the Principle of Linear Programming." The chapter culminates with a study of pull-back cones and establishes the following "universality" property of C[0,1]: every closed cone of a finite dimensional vector space is the pull-back cone of the cone of C[0,1] via a one-to-one operator from the space to C[0,1]. Chapter 4 investigates the fixed points and eigenvalues of an important class of positive operators known as Krein operators. A Krein space is an ordered Banach space having order units and a closed cone. A positive oper• ator T on a Krein space is a Krein operator if for any x > 0 the vector Tnx is an order unit for some n. Many integral operators are Krein operators. These operators possess some useful fixed points that are investigated in this chapter. Preface XI Chapters 5, 6, and 7 contain new material that, as far as we know, has not appeared before in any monograph. Chapter 5 develops in detail the theory of /C-lattices. An ordered vector space L is called a fC-lattice, where K is a super cone of L, i.e., K I) L+, if for every nonempty subset A of L the collection of all L+-upper bounds of A is nonempty and has a /C- infimum.
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