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Preface
in keeping with a long tradition in mathematical economics. We also ap-
proach the subject with our own personal presentiment that the special class
of Riesz spaces is somehow "perfect" and thus loosely conceive of general
ordered vector spaces as "deviations" from this "perfection." The book also
contains material that has not been published in a monograph form before.
The study of this material was initially motivated by various problems in
economics and econometrics.
The material is spread out in eight chapters. Chapter 8 is an Appendix
and contains some basic notions of functional analysis. Special attention is
paid to the properties of linear topologies and the separation of convex sets.
The results in this chapter (some of which are presented with proofs) are
used throughout the monograph without specific mention.
Chapter 1 presents the fundamental properties of wedges and cones.
Here we discuss Archimedean cones, lattice cones, extremal vectors of cones,
bases of cones, positive linear functionals and the important decomposabil-
ity property of cones known as the Riesz decomposition property. Chapter 2
introduces cones in topological vector spaces. This chapter illustrates the
variety of remarkable results that can be obtained when some link between
the order and the topology is imposed. The most important interrelationship
between a cone and a linear topology is known as normality. We discuss nor-
mal cones in detail and obtain several characterizations. In normed spaces,
the normality of the cone amounts to the norm boundedness of the order
intervals generated by the cone. In Chapter 2 we also introduce ideals and
present some of their useful order and topological properties.
Chapter 3 studies in detail cones in finite dimensional vector spaces. The
results here are much sharper. For instance, as we shall see, every closed cone
of a finite dimensional vector space is normal. The reader will find in this
chapter a study (together with a geometrical description) of the polyhedral
cones as well as a discussion of the properties of linear inequalities—including
a proof of "the Principle of Linear Programming." The chapter culminates
with a study of pull-back cones and establishes the following "universality"
property of C[0,1]: every closed cone of a finite dimensional vector space is
the pull-back cone of the cone of C[0,1] via a one-to-one operator from the
space to C[0,1].
Chapter 4 investigates the fixed points and eigenvalues of an important
class of positive operators known as Krein operators. A Krein space is an
ordered Banach space having order units and a closed cone. A positive oper-
ator T on a Krein space is a Krein operator if for any x 0 the vector
Tnx
is an order unit for some n. Many integral operators are Krein operators.
These operators possess some useful fixed points that are investigated in
this chapter.
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