**Mathematical Surveys and Monographs**

Volume: 105;
2003;
344 pp;
Hardcover

MSC: Primary 46; 47; 91;
Secondary 28

Print ISBN: 978-0-8218-3408-4

Product Code: SURV/105

List Price: $97.00

Individual Member Price: $77.60

**Electronic ISBN: 978-1-4704-1332-3
Product Code: SURV/105.E**

List Price: $97.00

Individual Member Price: $77.60

# Locally Solid Riesz Spaces with Applications to Economics: Second Edition

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*Charalambos D. Aliprantis; Owen Burkinshaw*

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

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.

#### Table of Contents

# Table of Contents

## Locally Solid Riesz Spaces with Applications to Economics: Second Edition

- Contents v6 free
- Preface of the 1st Edition vii8 free
- Preface of the 2nd Edition ix10 free
- List of Special Symbols xi12 free
- Chapter 1. The Lattice Structure of Riesz spaces 114 free
- 1.1. Elementary Properties of Riesz spaces 114
- 1.2. Ideals, Bands, and Riesz Subspaces 1023
- 1.3. Order Completeness and Projection Properties 1831
- 1.4. The Freudenthal Spectral Theorem 2538
- 1.5. The Main Inclusion Theorem 3144
- 1.6. Order Bounded Operators 3346
- 1.7. The Order Dual of a Riesz space 4154

- Chapter 2. Locally Solid Topologies 4962
- Chapter 3. Lebesgue Topologies 7588
- Chapter 4. Fatou Topologies 99112
- Chapter 5. Metrizability 119132
- Chapter 6. Weak Compactness in Riesz Spaces 143156
- Chapter 7. Lateral Completeness 179192
- Chapter 8. Market Economies 215228
- 8.1. Preferences and Utility Functions 215228
- 8.2. Exchange Economies and Efficiency 217230
- 8.3. Efficiency, Prices, and the Welfare Theorems 221234
- 8.4. Properness 224237
- 8.5. Properness and Efficiency 228241
- 8.6. Equilibrium 231244
- 8.7. Continuity Properties of Supporting Prices 233246
- 8.8. The Utility Space of an Economy and Efficiency 236249
- 8.9. Existence of Equilibria 240253
- 8.10. The Core of an Economy 243256
- 8.11. Replication 245258
- 8.12. Edgeworth Equilibria 247260
- 8.13. Core Equivalence 251264
- 8.14. The Single Sector Growth Model 257270

- Chapter 9. Solutions to the Exercises 267280
- 9.1. Chapter 1: The Lattice Structure of Riesz Spaces 267280
- 9.2. Chapter 2: Locally Solid Topologies 279292
- 9.3. Chapter 3: Lebesgue Topologies 284297
- 9.4. Chapter 4: Fatou Topologies 289302
- 9.5. Chapter 5: Metrizability 295308
- 9.6. Chapter 6: Weak Compactness in Riesz Spaces 301314
- 9.7. Chapter 7: Lateral Completeness 307320
- 9.8. Chapter 8: Market Economies 317330

- Bibliography 331344
- Index 337350

#### Readership

Graduate students and research mathematicians interested in functional analysis and applications to economics; scientists and engineers interested in order structures.

#### Reviews

Self-contained … should be accessible to any student with a standard course in functional analysis.

-- Mathematical Reviews

The authors as well as the Publisher are to be commended for having enriched the mathematical literature in this area of research with this beautifully written new edition of the original book.

-- Zentralblatt MATH