目錄

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