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141 Graham Jameson University of Warwick, Coventry/England

Ordered Linear Spaces

Springer-Verlag Berlin · Heidelberg · New York 1970

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is c:oncemed, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying of similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the pablisher, the amouat of the fee to bedetermined by a,veement with the publisher.

by Springer-Verlag Birlin . Hddelbetg 1970. Library of Congress.CatalQ8 Card Number 7H2j282. Printed in Getmarly_ Tide No. 3291

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PREFACE Most of the familiar real linear spaces have a natural ordering. The study of ordered linear spaces was initiated in the late 1930's (by Riesz, Freudenthal, Kantorovic, Kakutani and others), since when a large and scattered research literature haa grown up. an attempt to

a

This work

balanced introductory treatment of the various

branches of the subject, without giving an encyclopaedic survey of any of them. text.

Suggestions for further reading are distributed through the These do not constitute a complete bibliography of the topics

considered, but it is hoped that they will be found sufficient to launch the reader into the relevant literature.

The treatment of the elemen-

tary theory is quite detailed, because it is the author's belief that a good mastery of the basic material is a necessary condition for successful "advanced" work in any SUbject.

The more elementary sections,

in fact, formed the basis of an undergraduate course given at the of Warwick in 1968. Chapters land 2 deal with the purely algebraic theory, while Chapters 3 and 4 are concerned with ordered topological linear spaces. Chapter 3 is independent of Chapter 2. ordered algebras is designed to give a treatment, of this subject.

A short final chapter on rather than a systematic

However, this appears to be the natural

context for certain results on monotonic linear mappings without which any account of ordered linear spaces would be incomplete. An introductory Chapter 0 is included to summarise the terminology used concerning orderings and linear spaces (Where alternatives exist), and the main results to be assumed known. facilitate reference.

The intention is purely to

Chapter 0 is neither self­contained (e.g. we

do not repeat elementary definitions such as "topological linear space"), nor in any way a balanced summary of the SUbjects concerned. No attempt is made to consider linear spaces over fields other than the real numbers, but some care is taken to point out which results

iv require no scalar field at all, and therefore apply to ordered commutative groups.

To preserve conceptual simplicity, however, we ignore

any possible generalisations to non­commutative groups.

For a detailed

treatment of ordered groups, the reader is referred to Fuchs (1). The choice of material is to some extent