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' bookLocally 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 operatorsbetween 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 theexistence 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 presentscomplete 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.

Riesz spaces are named after Frigyes Riesz who first defined them in his 1928 paper "Sur la décomposition des opérations fonctionelles linéaires." Most early work on Riesz spaces addressed the algebraic side of the theory, ignoring the analytic implications, but exploring normed Riesz spaces and Banach lattices. This book concentrates on the general theory of topological Riesz spaces, with a unified approach to locally solid Riesz spaces, emphasizing the relationship between the order structure and the topological structure.

This monograph is a systematic exposition of the authors' research on general equi librium models with an infinite number of commodities. It is intended to serve both as a graduate text on aspects of general equilibrium theory and as an introduction, for economists and mathematicians working in mathematical economics, to current research in a frontier area of general equilibrium theory. To this end, we have pro vided two introductory chapters on the basic economic model and the mathematical framework. The exercises at the end of each section complement the main exposition. Chapter one is a concise but substantiative discussion of the questions of exis tence and optimality of competitive equilibria in the Walrasian general equilibrium model of an economy with a finite number of households, firms and commodities. Our extension of this model to economies with an infinite number of commodities constitutes the core material of this book and begins in chapter three. Readers fa miliar with the Walrasian general equilibrium model as exposited in (13], [23] or [52] may treat chapter one as a handy reference for the main economic concepts and notions that are used throughout the book.

Reprinted by popular demand, this monograph presents a comprehensive study of positive operators between Riesz spaces and Banach lattices. Since publication of this book in 1985, the subject of positive operators and Riesz spaces has found practical applications in disciplines including social sciences and engineering. This book examines positive operators in the setting of Riesz spaces and Banach lattices, from both the algebraic and topological points of view.

This text was born out of an advanced mathematical economics seminar at Caltech in 1989-90. We realized that the typical graduate student in mathematical economics has to be familiar with a vast amount of material that spans several traditional fields in mathematics. Much of the mate rial appears only in esoteric research monographs that are designed for specialists, not for the sort of generalist that our students need be. We hope that in a small way this text will make the material here accessible to a much broader audience. While our motivation is to present and orga nize the analytical foundations underlying modern economics and finance, this is a book of mathematics, not of economics. We mention applications to economics but present very few of them. They are there to convince economists that the material has so me relevance and to let mathematicians know that there are areas of application for these results. We feel that this text could be used for a course in analysis that would benefit math ematicians, engineers, and scientists. Most of the material we present is available elsewhere, but is scattered throughout a variety of sources and occasionally buried in obscurity. Some of our results are original (or more likely, independent rediscoveries). We have included some material that we cannot honestly say is neces sary to understand modern economic theory, but may yet prove useful in future research.

This monograph develops the theory of pre-Riesz spaces, which are the partially ordered vector spaces that embed order densely into Riesz spaces. Concepts from Riesz space theory such as disjointness, ideals, and bands are extended to pre-Riesz spaces. The analysis revolves around embedding techniques, including the Riesz completion and the functional representation. In the same spirit, norms and topologies on a pre-Riesz space and their extensions to the Riesz completion are examined. The generalized concepts are used to investigate disjointness preserving operators on pre-Riesz spaces and related notions. The monograph presents recent results as well as being an accessible introduction to the theory of partially ordered vector spaces and positive operators. Contents A primer on ordered vector spaces Embeddings, covers, and completions Seminorms on pre-Riesz spaces Disjointness, bands, and ideals in pre-Riesz spaces Operators on pre-Riesz spaces

This book exclusively deals with the study of almost convergence and statistical convergence of double sequences. The notion of “almost convergence” is perhaps the most useful notion in order to obtain a weak limit of a bounded non-convergent sequence. There is another notion of convergence known as the “statistical convergence”, introduced by H. Fast, which is an extension of the usual concept of sequential limits. This concept arises as an example of “convergence in density” which is also studied as a summability method. Even unbounded sequences can be dealt with by using this method. The book also discusses the applications of these non-matrix methods in approximation theory. Written in a self-contained style, the book discusses in detail the methods of almost convergence and statistical convergence for double sequences along with applications and suitable examples. The last chapter is devoted to the study convergence of double series and describes various convergence tests analogous to those of single sequences. In addition to applications in approximation theory, the results are expected to find application in many other areas of pure and applied mathematics such as mathematical analysis, probability, fixed point theory and statistics.

Measure Theory has played an important part in the development of functional analysis: it has been the source of many examples for functional analysis, including some which have been leading cases for major advances in the general theory, and certain results in measure theory have been applied to prove general results in analysis. Often the ordinary functional analyst finds the language and a style of measure theory a stumbling block to a full understanding of these developments. Dr Fremlin's aim in writing this book is therefore to identify those concepts in measure theory which are most relevant to functional analysis and to integrate them into functional analysis in a way consistent with that subject's structure and habits of thought. This is achieved by approaching measure theory through the properties of Riesz spaces and especially topological Riesz spaces. Thus this book gathers together material which is not readily available elsewhere in a single collection and presents it in a form accessible to the first-year graduate student, whose knowledge of measure theory need not have progressed beyond that of the ordinary lebesgue integral.

Functional Analysis has become one of the main branches in Chinese mathematics. Many outstanding contributions and results have been achieved over the past sixty years. This authoritative collection is complementary to Western studies in this field, and seeks to summarise and introduce the historical progress of the development of Functional Analysis in China from the 1940s to the present. A broad range of topics is covered, such as nonlinear functional analysis, linear operator theory, theory of operator algebras, applications including the solvability of some partial differential equations, and special spaces that contain Banach spaces and topological vector spaces. Some of these papers have made a significant impact on the mathematical community worldwide. Audience: This volume will be of interest to mathematicians, physicists and engineers at postgraduate level.