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.
Author(s): Charalambos D. Aliprantis, Owen Burkinshaw
Series: Mathematical Surveys and Monographs, 105
Edition: 2
Publisher: American Mathematical Society
Year: 2003
Language: English
Pages: 360
Front Cover......Page 1
Title Page......Page 2
Copyright Information......Page 3
Dedication......Page 4
Contents......Page 6
Preface of the 1st Edition......Page 8
Preface of the 2nd Edition......Page 10
List of Special Symbols......Page 12
1.1. Elementary Properties of Riesz spaces......Page 14
1.2. Ideals, Bands, and Riesz Subspaces......Page 23
1.3. Order Completeness and Projection Properties......Page 31
1.4. The Freudenthal Spectral Theorem......Page 38
1.5. The Main Inclusion Theorem......Page 44
1.6. Order Bounded Operators......Page 46
1.7. The Order Dual of a Riesz space......Page 54
2.1. Linear Topologies on Vector Spaces......Page 62
2.2. The Basic Properties of Locally Solid Topologies......Page 68
2.3. Locally Convex-solid Topologies......Page 72
2.4. Topological Completion of a Locally Solid Riesz Space......Page 79
3.1. Examples and Properties of Lebesgue Topologies......Page 88
3.2. Locally Convex-solid Lebesgue Topologies......Page 93
3.3. Lebesgue Properties and Lp-Spaces......Page 98
4.1. Basic Properties of the Fatou Topologies......Page 112
4.2. The Structure of the Fatou Topologies......Page 115
4.3. Topological Completeness and Fatou Topologies......Page 121
4.4. Quotient Riesz Spaces and Fatou Properties......Page 127
5.1. Upper and Lower Elements......Page 132
5.2. Frchet Topologies......Page 138
5.3. The Pseudo Lebesgue Properties......Page 142
5.4. The B-Property......Page 148
5.5. Comparing Locally Solid Topologies......Page 149
6.1. Topologies on the Duals of a Riesz Space......Page 156
6.2. Weak Compactness in the Order Duals......Page 161
6.3. Weak Sequential Convergence......Page 171
6.4. Compact Solid Sets......Page 175
6.5. Semireflexive Riesz Spaces......Page 186
7.1. Laterally Complete Riesz Spaces......Page 192
7.2. The Universal Completion......Page 200
7.3. Lateral and Universal Completeness......Page 209
7.4. Lateral Completeness and Local Solidness......Page 214
7.5. Minimal Locally Solid Topologies......Page 220
8.1. Preferences and Utility Functions......Page 228
8.2. Exchange Economies and Efciency......Page 230
8.3. Efciency, Prices, and the Welfare Theorems......Page 234
8.4. Properness......Page 237
8.5. Properness and Effciency......Page 241
8.6. Equilibrium......Page 244
8.7. Continuity Properties of Supporting Prices......Page 246
8.8. The Utility Space of an Economy and Efciency......Page 249
8.9. Existence of Equilibria......Page 253
8.10. The Core of an Economy......Page 256
8.11. Replication......Page 258
8.12. Edgeworth Equilibria......Page 260
8.13. Core Equivalence......Page 264
8.14. The Single Sector Growth Model......Page 270
9.1. Chapter 1: The Lattice Structure of Riesz Spaces......Page 280
9.2. Chapter 2: Locally Solid Topologies......Page 292
9.3. Chapter 3: Lebesgue Topologies......Page 297
9.4. Chapter 4: Fatou Topologies......Page 302
9.5. Chapter 5: Metrizability......Page 308
9.6. Chapter 6: Weak Compactness in Riesz Spaces......Page 314
9.7. Chapter 7: Lateral Completeness......Page 320
9.8. Chapter 8: Market Economies......Page 330
Bibliography......Page 344
Index......Page 350
Back Cover......Page 360
