Author | Schaefer, H. H. author |
---|---|
Title | Topological Vector Spaces [electronic resource] / by H. H. Schaefer, M. P. Wolff |
Imprint | New York, NY : Springer New York : Imprint: Springer, 1999 |
Edition | Second Edition |
Connect to | http://dx.doi.org/10.1007/978-1-4612-1468-7 |
Descript | XII, 349 p. online resource |
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 Banachโs 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-Dieudonnรฉ, and Krein-ล 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 Vector Spaces 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*โand W*โAlgebras -- 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. Kaplanskyโs Density Theorem -- 8 Projections and Types of W*-Algebras -- Exercises -- Appendix. Spectral Properties of Positive Operators -- 1 Elementary Properties of the Resolvent -- 2 Pringsheimโs Theorem and Its Consequences -- 3 The Peripheral Point Spectrum -- Index of Symbols