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Topological Vector Spaces

The present book is intended to be a systematic text on topological vector spaces and presupposes familiarity with the elements of general topology and linear algebra. The author has found it unnecessary to rederive these results, since they are equally basic for many other areas of mathematics, and...

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Bibliographic Details
Main Author: Schaefer, H.H.
Format: eBook
Language:English
Published: New York, NY Springer New York 1999, 1999
Edition:2nd ed. 1999
Series:Graduate Texts in Mathematics
Subjects:
Mathematical Analysis
Analysis
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 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’s Homomorphism Theorem
  • 3 Spaces of Linear Mappings
  • 4 Equicontinuity. The Principle of Uniform Boundedness and the Banach-Steinhaus Theorem
  • 5 Bilinear Mappings
  • 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*—and W*—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’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’s Theorem and Its Consequences
  • 3 The Peripheral Point Spectrum
  • Index of Symbols
  • 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-Dieudonné, and Krein-Šmulian
  • 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

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