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. 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 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.
