Locally Solid Riesz Spaces with Applications to Economics (Mathematical Surverys and Monographs Series) / Edition 2

Locally Solid Riesz Spaces with Applications to Economics (Mathematical Surverys and Monographs Series) / Edition 2

by Charalambos D. Aliprantis, Owen Burkinshaw, Owen Burkinshaw
     
 

ISBN-10: 0821834088

ISBN-13: 9780821834084

Pub. Date: 11/01/2003

Publisher: American Mathematical Society

Riesz space (or a vector lattice) is an ordered vector space that is simultaneously a lattice. A topological Riesz space (also called a locally solid Riesz space) is a Riesz space equipped with a linear topology that has a base consisting of solid sets. Riesz spaces and ordered vector spaces play an important role in analysis and optimization. They also provide the

Overview

Riesz space (or a vector lattice) is an ordered vector space that is simultaneously a lattice. A topological Riesz space (also called a locally solid Riesz space) is a Riesz space equipped with a linear topology that has a base consisting of solid sets. Riesz spaces and ordered vector spaces play an important role in analysis and optimization. They also provide the natural framework for any modern theory of integration. This monograph is the revised edition of the authors' book Locally Solid Riesz Spaces (1978, Academic Press). It presents an extensive and detailed study (with complete proofs) of topological Riesz spaces. The book starts with a comprehensive exposition of the algebraic and lattice properties of Riesz spaces and the basic properties of order bounded operators between Riesz spaces. Subsequently, it introduces and studies locally solid topologies on Riesz spaces— the main link between order and topology used in this monograph. Special attention is paid to several continuity properties relating the order and topological structures of Riesz spaces, the most important of which are the Lebesgue and Fatou properties. A new chapter presents some surprising applications of topological Riesz spaces to economics. In particular, it demonstrates that the existence of economic equilibria and the supportability of optimal allocations by prices in the classical economic models can be proven easily using techniques from the theory of topological Riesz spaces. At the end of each chapter there are exercises that complement and supplement the material in the chapter. The last chapter of the book presents complete solutions to all exercises. Prerequisites are the fundamentals of real analysis, measure theory, and functional analysis. This monograph will be useful to researchers and graduate students in mathematics. It will also be an important reference tool to mathematical economists and to all scientists and engineers who use order structures in their research.

Product Details

ISBN-13:
9780821834084
Publisher:
American Mathematical Society
Publication date:
11/01/2003
Series:
Mathematical Surveys and Monographs, #105
Pages:
360
Product dimensions:
7.20(w) x 10.28(h) x 0.91(d)

Table of Contents

Preface of the 1st Editionvii
Preface of the 2nd Editionix
List of Special Symbolsxi
Chapter 1.The Lattice Structure of Riesz spaces1
1.1.Elementary Properties of Riesz spaces1
1.2.Ideals, Bands, and Riesz Subspaces10
1.3.Order Completeness and Projection Properties18
1.4.The Freudenthal Spectral Theorem25
1.5.The Main Inclusion Theorem31
1.6.Order Bounded Operators33
1.7.The Order Dual of a Riesz space41
Chapter 2.Locally Solid Topologies49
2.1.Linear Topologies on Vector Spaces49
2.2.The Basic Properties of Locally Solid Topologies55
2.3.Locally Convex-solid Topologies59
2.4.Topological Completion of a Locally Solid Riesz Space66
Chapter 3.Lebesgue Topologies75
3.1.Examples and Properties of Lebesgue Topologies75
3.2.Locally Convex-solid Lebesgue Topologies80
3.3.Lebesgue Properties and L[subscript p]-Spaces85
Chapter 4.Fatou Topologies99
4.1.Basic Properties of the Fatou Topologies99
4.2.The Structure of the Fatou Topologies102
4.3.Topological Completeness and Fatou Topologies108
4.4.Quotient Riesz Spaces and Fatou Properties114
Chapter 5.Metrizability119
5.1.Upper and Lower Elements119
5.2.Frechet Topologies125
5.3.The Pseudo Lebesgue Properties129
5.4.The B-Property135
5.5.Comparing Locally Solid Topologies136
Chapter 6.Weak Compactness in Riesz Spaces143
6.1.Topologies on the Duals of a Riesz Space143
6.2.Weak Compactness in the Order Duals148
6.3.Weak Sequential Convergence158
6.4.Compact Solid Sets162
6.5.Semireflexive Riesz Spaces173
Chapter 7.Lateral Completeness179
7.1.Laterally Complete Riesz Spaces179
7.2.The Universal Completion187
7.3.Lateral and Universal Completeness196
7.4.Lateral Completeness and Local Solidness201
7.5.Minimal Locally Solid Topologies207
Chapter 8.Market Economies215
8.1.Preferences and Utility Functions215
8.2.Exchange Economies and Efficiency217
8.3.Efficiency, Prices, and the Welfare Theorems221
8.4.Properness224
8.5.Properness and Efficiency228
8.6.Equilibrium231
8.7.Continuity Properties of Supporting Prices233
8.8.The Utility Space of an Economy and Efficiency236
8.9.Existence of Equilibria240
8.10.The Core of an Economy243
8.11.Replication245
8.12.Edgeworth Equilibria247
8.13.Core Equivalence251
8.14.The Single Sector Growth Model257
Chapter 9.Solutions to the Exercises267
9.1.Chapter 1: The Lattice Structure of Riesz Spaces267
9.2.Chapter 2: Locally Solid Topologies279
9.3.Chapter 3: Lebesgue Topologies284
9.4.Chapter 4: Fatou Topologies289
9.5.Chapter 5: Metrizability295
9.6.Chapter 6: Weak Compactness in Riesz Spaces301
9.7.Chapter 7: Lateral Completeness307
9.8.Chapter 8: Market Economies317
Bibliography331
Index337

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