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
Locally Solid Riesz Spaces (1978, Academic Press). It
presents an extensive and detailed study (with complete proofs) of
topological Riesz spaces. The book starts with a comprehensive
exposition of the algebraic and lattice properties of Riesz spaces and
the basic properties of order bounded operators between Riesz
spaces. Subsequently, it introduces and studies locally solid
topologies on Riesz spaces— the main link between order and
topology used in this monograph. Special attention is paid to several
continuity properties relating the order and topological structures of
Riesz spaces, the most important of which are the Lebesgue and Fatou
properties. A new chapter presents some surprising applications of
topological Riesz spaces to economics. In particular, it demonstrates
that the existence of economic equilibria and the supportability of
optimal allocations by prices in the classical economic models can be
proven easily using techniques from the theory of topological Riesz
spaces.
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.
Readership
Graduate students and research mathematicians interested in
functional analysis and applications to economics; scientists and engineers
interested in order structures.