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WorldCat.org

Prerequisites.- A. Sets and Order.- B. General Topology.- C. Linear Algebra.- I. Topological Vector Spaces.- 1 Vector Space Topologies.- 2 Product Spaces, Subspaces, Direct Sums, Quotient Spaces.- 3 Topological Vector Spaces of Finite Dimension.- 4 Linear Manifolds and Hyperplanes.- 5 Bounded Sets.- 6 Metrizability.- 7 Complexification.- Exercises.- II. Locally Convex Topological Vector Spaces.- 1 Convex Sets and Semi-Norms.- 2 Normed and Normable Spaces.- 3 The Hahn-Banach Theorem.- 4 Locally Convex Spaces.- 5 Projective Topologies.- 6 Inductive Topologies.- 7 Barreled Spaces.- 8 Bornological Spaces.- 9 Separation of Convex Sets.- 10 Compact Convex Sets.- Exercises.- III. Linear Mappings.- 1 Continuous Linear Maps and Topological Homomorphisms.- 2 Banach&rsquo;s Homomorphism Theorem.- 3 Spaces of Linear Mappings.- 4 Equicontinuity. The Principle of Uniform Boundedness and the Banach-Steinhaus Theorem.- 5 Bilinear Mappings.- 6 Topological Tensor Products.- 7 Nuclear Mappings and Spaces.- 8 Examples of Nuclear Spaces.- 9 The Approximation Property. Compact Maps.- Exercises.- IV. Duality.- 1 Dual Systems and Weak Topologies.- 2 Elementary Properties of Adjoint Maps.- 3 Locally Convex Topologies Consistent with a Given Duality.The Mackey-Arens Theorem.- 4 Duality of Projective and Inductive Topologies.- 5 Strong Dual of a Locally Convex Space. Bidual. Reflexive Spaces.- 6 Dual Characterization of Completeness. Metrizable Spaces. Theorems of Grothendieck, Banach-Dieudonn&eacute;, and Krein-&Scaron;mulian.- 7 Adjoints of Closed Linear Mappings.- 8 The General Open Mapping and Closed Graph Theorems.- 9 Tensor Products and Nuclear Spaces.- 10 Nuclear Spaces and Absolute Summability.- 11 Weak Compactness. Theorems of Eberlein and Krein.- Exercises.- V. Order Structures.- 1 Ordered VectorSpaces over the Real Field.- 2 Ordered Vector Spaces over the Complex Field.- 3 Duality of Convex Cones.- 4 Ordered Topological Vector Spaces.- 5 Positive Linear Forms and Mappings.- 6 The Order Topology.- 7 Topological Vector Lattices.- 8 Continuous Functions on a Compact Space. Theorems of Stone-Weierstrass and Kakutani.- Exercises.- VI. C*&mdash;and W*&mdash;Algebras.- 1 Preliminaries.- 2 C*-Algebras.The Gelfand Theorem.- 3 Order Structure of a C*-Algebra.- 4 Positive Linear Forms. Representations.- 5 Projections and Extreme Points.- 6 W*-Algebras.- 7 Von Neumann Algebras. Kaplansky&rsquo;s Density Theorem.- 8 Projections and Types of W*-Algebras.- Exercises.- Appendix. Spectral Properties of Positive Operators.- 1 Elementary Properties of the Resolvent.- 2 Pringsheim&rsquo;s Theorem and Its Consequences.- 3 The Peripheral Point Spectrum.- Index of Symbols.

This book is intended to be a systematic text on topological vector spaces and presupposes familiarity with the elements of general topology and linear algebra. Similarly, the elementary facts on Hilbert and Banach spaces are widely known and are not discussed in detail in this book, which is mainly addressed to those readers who have attained and wish to get beyond the introductory level. Each of the chapters is preceded by an introduction and followed by exercises. These exercises are devoted to further results and supplements, in particular, to examples and counter-examples. Hints have been given where it seemed appropriate. This second edition has been thoroughly revised and includes a new chapter on C * and W * algebras

Topological Vector Spaces