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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.

Measure theory. --- Riesz spaces.

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Functional analysis --- Espaces vectoriels topologiques ordonnés. --- Linear topological spaces, Ordered --- Riesz spaces --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Mathematical Theory --- Riesz vector spaces --- Vector lattices --- Lattice theory --- Vector spaces --- Riesz spaces. --- Espaces vectoriels topologiques ordonnés.

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Functional analysis --- Banach spaces. --- Partially ordered spaces. --- Riesz spaces. --- Locally convex spaces. --- Banach, Espaces de --- Espaces partiellement ordonnés --- Riesz, Espaces de --- Espaces localement convexes --- Banach spaces --- Partially ordered spaces --- Riesz spaces --- Locally convex spaces --- Espaces partiellement ordonnés --- Analyse fonctionnelle --- Espaces vectoriels topologiques ordonnés --- Linear topological spaces, Ordered --- Espaces vectoriels topologiques ordonnés. --- Functional analysis. --- Linear topological spaces, Ordered. --- Espaces vectoriels ordonnes

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The book deals with the structure of vector lattices, i.e. Riesz spaces, and Banach lattices, as well as with operators in these spaces. The methods used are kept as simple as possible. Almost no prior knowledge of functional analysis is required. For most applications some familiarity with the oridinary Lebesgue integral is already sufficient. In this respect the book differs from other books on the subject. In most books on functional analysis (even excellent ones) Riesz spaces, Banach lattices and positive operators are mentioned only briefly, or even not at all. The present book shows how these subjects can be treated without undue extra effort. Many of the results in the book were not yet known thirty years ago; some even were not known ten years ago.

Analytical spaces --- Riesz spaces. --- Operator theory. --- Functional analysis. --- Functional Analysis. --- Operator Theory. --- Functional analysis --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Riesz vector spaces --- Vector lattices --- Lattice theory --- Vector spaces

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Most classes of operators that are not isomorphic embeddings are characterized by some kind of a "smallness" condition. Narrow operators are those operators defined on function spaces that are "small" at {-1,0,1}-valued functions, e.g. compact operators are narrow. The original motivation to consider such operators came from theory of embeddings of Banach spaces, but since then they were also applied to the study of the Daugavet property and to other geometrical problems of functional analysis. The question of when a sum of two narrow operators is narrow, has led to deep developments of the theory of narrow operators, including an extension of the notion to vector lattices and investigations of connections to regular operators. Narrow operators were a subject of numerous investigations during the last 30 years. This monograph provides a comprehensive presentation putting them in context of modern theory. It gives an in depth systematic exposition of concepts related to and influenced by narrow operators, starting from basic results and building up to most recent developments. The authors include a complete bibliography and many attractive open problems.

Narrow operators. --- Riesz spaces. --- Function spaces. --- Spaces, Function --- Functional analysis --- Riesz vector spaces --- Vector lattices --- Lattice theory --- Vector spaces --- Operators, Narrow --- Operator theory --- Function Space. --- Narrow Operator. --- Vector Lattice.