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**Locally solid Riesz spaces with applications to economics.
2nd ed.**
*(English)*
Zbl 1043.46003

Mathematical Surveys and Monographs 105. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3408-8/hbk). xii, 344 p. (2003).

The first edition of the book under review appeared in 1978 (Academic Press, New York, see Zbl 0402.46005). At that time, it was the first book exclusively devoted to the growing subject of the theory of locally solid linear topological Riesz spaces. Since its appearance, the theory of locally solid linear topologies has seen many new developments and applications. With respect to the latter issue, the authors showed that the theory of locally solid Riesz spaces can be fruitfully applied to certain aspects of economic theory.

The new edition, in addition to bringing the theory of locally solid Riesz spaces up-to-date, has been enriched with a new chapter of more than fifty pages that presents a self-contained and beautifully written exposition of the main results of equilibria in markets in the Arrow-Debreu-McKenzie framework in the language of the theory of locally solid topological Riesz spaces. This material is distributed over fourteen sections with the following titles, listed here in the order they appear: Preferences and Utility Functions; Exchange Economies and Efficiency; Efficiency, Prices, and the Welfare Theorems; Properness; Properness and Efficiency; Equilibrium; Continuity Properties of Supporting Prices; The Utility Space of an Economy and Efficiency; Existence of Equilibria; The Core of an Economy; Replication; Edgeworth Equilibria; Core Equivalence; The Single Sector Growth Model.

Another new chapter is devoted to the presentation of all solutions of the exercises. The bibliography is brought up-to-date as well and contains 132 references to the relevant literature.

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.

The new edition, in addition to bringing the theory of locally solid Riesz spaces up-to-date, has been enriched with a new chapter of more than fifty pages that presents a self-contained and beautifully written exposition of the main results of equilibria in markets in the Arrow-Debreu-McKenzie framework in the language of the theory of locally solid topological Riesz spaces. This material is distributed over fourteen sections with the following titles, listed here in the order they appear: Preferences and Utility Functions; Exchange Economies and Efficiency; Efficiency, Prices, and the Welfare Theorems; Properness; Properness and Efficiency; Equilibrium; Continuity Properties of Supporting Prices; The Utility Space of an Economy and Efficiency; Existence of Equilibria; The Core of an Economy; Replication; Edgeworth Equilibria; Core Equivalence; The Single Sector Growth Model.

Another new chapter is devoted to the presentation of all solutions of the exercises. The bibliography is brought up-to-date as well and contains 132 references to the relevant literature.

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.

Reviewer: W. A. J. Luxemburg (Pasadena)

### MSC:

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

46A40 | Ordered topological linear spaces, vector lattices |

46A20 | Duality theory for topological vector spaces |

46A04 | Locally convex FrĂ©chet spaces and (DF)-spaces |