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Lecture Notes in
Mathematics
A collection of informal reports and seminars
Edited by A. Dold, Heidelberg and B. Eckmann, ZOrich
Series: Mathematics Institute, University of Warwick
Adviser: D. B. A. Epstein
141
Graham Jameson
University of Warwick, Coventry/England
Ordered Linear Spaces
$
z.E,=
Springer-Verlag
Berlin.Heidelberg. New York 1970
This work is subject to copyright. All rights are reCerved, whether the whole or part of the material is ¢onzet'ned,
specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine
o~" similar means, and storage in data hanks.
Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher,
the amount of the fee to be determined by agreement with the publiSher.
© by Springer-Verlag Borlin • FIcid~lber~ 1970. Library of Congress Catalq~g Card Number 73-125282. Printed in Gergtany~
Tide No. 3297
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, Kantorovi6, Kakutani and others), since when
a large and scattered research literature has grown up. This work is
an attempt to provide a balanced introductory treatment of the various
branches of the subject, without giving an encyclopaedic survey of any
of them. Suggestions for further reading are distributed through the
text. 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 suc-
cessful "advanced" work in any subject. The more elementary sections,
in fact, formed the basis of an undergraduate course given at the
University of Warwick in 1968.
Chapters i and 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. A short final chapter on
ordered algebras is designed to give a tas~ rather than a systematic
treatment, of this subject. 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 concerningorderings and linear spaces (where alternatives exist),
and the main results to be assumed known. The intention is purely to
facilita~cereference. Chapter 0 is neither self-contained (e.g. we
do not repeat elementary definitions such as "topological linear space"),
nor in any way a balanced s~mmary 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 commut-
ative groups. To preserve conceptual simplicity, however, we ignore
any possible generalisations to non-commutative groups. For a detailed
treatment of ordered groups, the reader is referred to Fuchs (1).
The choice of material is to some extent complementary to that in
the book of Peressinl (2). For example, more space is given to
ordered normed spaces. The formulation and proof of 1.7.3 (possibly
the most important theorem) are rather different from previous versions.
A similar comment applies to the main theorem on extremal monotonic
linear functionals (1.8.2), together with its topological version
(3.1.12). Some of the material on cones with bases (sections 3.8 and
4.4.4) is new, at least in presentation.
The author is indebted to Prof. F. F. Bonsall, without whose
constant advice and encouragement these notes would never have reached
completion, and to Mrs Susan Elworthy for typing the text.
0.I,
0.2.
0.3.
I.I.
1.2.
1.3.
1.4.
1.5.
1.6.
1.7.
1.8.
1.9.
I.i0.
2,1.
2.2.
2.3.
2.4.
CONTENTS
CHAPTER
0 :
PRELI~INARIES
Order i ngs .......................................
Linear spaces ...................................
Topological linear spaces .......................
CHAPTER i: ORDERED LINEAR SPACES
Introduction ....................................
Wedges, cones and orderings .....................
Order-convexity .................................
0rder-units and Archimedean orderings ...........
Direct sums and quotient spaces .................
Linear mappings and functionals .................
Extension and separation theorems ...............
Decomposition of a linear functional into
monotonic components .......................
Extremal monotonic linear functionals ............
Bases for cones .................................
Regular and maximal order-convex subspaces .......
CHAPTER 2: LINEAR LATTICES AND RIESZ SPACES
Introduction ....................................
Riesz spaces ....................................
Linear lattices: basic theory ..................
Positive and negative parts and moduli ...........
Disjointness and bands ..........................
viii
X
xii
1
2
9
iI
16
18
24
27
31
34
37
4o
43
51
57
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