| Preface Preface to the Second Edition CHAPTER I Hilbert Spaces 1.Elementary Properties and Examples 2.Orthogonality 3.The Riesz Representation Theorem 4.Orthonormal Sets of Vectors and Bases 5.Isomorphic Hilbert Spaces and the Fourier Transform for the Circle 6.The Direct Sum of Hilbert Spaces CHAPTER Ⅱ Operators on Hilbert Space 1.Elementary Properties and Examples 2.The Adjoint of an Operator 3.Projections and Idempotents; Invariant and Reducing Subspaces 4.Compact Operators 5.* The Diagonalization of Compact Self-Adjoint Operators 6.* An Application: Sturm-Liouville Systems 7.* The Spectral Theorem and Functional Calculus for Compact Normal Operators 8.* Unitary Equivalence for Compact Normal Operators CHAPTER Ⅲ Banach Spaces 1.Elementary Properties and Examples 2.Linear Operators on Normed Spaces 3.Finite Dimensional Normed Spaces 4.Quotients and Products of Normed Spaces 5.Linear Functionals 6.The Hahn-Banach Theorem 7.* An Application: Banach Limits 8.* An Application: Runge's Theorem 9.* An Application: Ordered Vector Spaces 1.The Dual of a Quotient Space and a Subspace 11.Reflexive Spaces 12.The Open Mapping and Closed Graph Theorems 13.Complemented Subspaces of a Banach Space 14.The Principle of Uniform Boundedness CHAPTER IV Locally Convex Spaces 1.Elementary Properties and Examples 2.Metrizable and Normable Locally Convex Spaces 3.Some Geometric Consequences of the Hahn-Banach Theorem 4.* Some Examples of the Dual Space of a Locally Convex Space 5.* Inductive Limits and the Space of Distributions CHAPTER V Weak Topologies 1.Duality 2.The Dual of a Subspace and a Quotient Space 3.Alaoglu's Theorem 4.Reflexivity Revisited 5.Separability and Metrizability 6.* An Application: The Stone-t ech Compactification 7.The Krein-Milman Theorem 8.An Application: The Stone-Weierstrass Theorem 9.* The Schauder Fixed Point Theorem 10.* The Ryll-Nardzewski Fixed Point Theorem 11.* An Application: Haar Measure on a Compact Group 12.* The Krein-Smulian Theorem 13.* Weak Compactness CHAPTER VI Linear Operators on a Banach Space CHAPTER VII Banach Agebras and Spectral Theory for Operators on a Banach Space CHAPTER VIII C-Algebras CHAPTER IX Normal Operators on Hilbert Space CHAPTER X Unbounded Operators CHAPTER XI Fredholm Theory APPENDIX A APPENDIX B APPENDIX C Bibliography List of Symbols Index |
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