A Course in Functional Analysis

Cover
Title
Preface
Preface to the Second Edition
Contents
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
10. 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-Cech 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
1. The Adjoint of a Linear Operator
2.* The Banach-Stone Theorem
3. Compact Operators
4. Invariant Subspaces
5. Weakly Compact Operators
CHAPTER VII Banach Algebras and Spectral Theory for Operators on a Banach Space
1. Elementary Properties and Examples
2. Ideals and Quotients
3. The Spectrum
4. The Riesz Functional Calculus
5. Dependence of the Spectrum on the Algebra
6. The Spectrum of a Linear Operator
7. The Spectral Theory of a Compact Operator
8. Abelian Banach Algebras
9.* The Group Algebra of a Locally Compact Abelian Group
CHAPTER VIII C*-Algebras
1. Elementary Properties and Examples
2. Abelian C*-Algebras and the Functional Calculus in C*-Algebras
3. The Positive Elements in a C*-Algebra
4.* Ideals and Quotients of C*-Algebras
5.* Representations of C*-Algebras and the Gelfand-Naimark-Segal Construction
CHAPTER IX Normal Operators on Hilbert Space
1. Spectral Measures and Representations of Abelian C*-Algebras
2. The Spectral Theorem
3. Star-Cyclic Normal Operators
4. Some Applications of the Spectral Theorem
5. Topologies on B(K)
6. Commuting Operators
7. Abelian von Neumann Algebras
8. The Functional Calculus for Normal Operators: The Conclusion of the Saga
9. Invariant Subspaces for Normal Operators
10. Multiplicity Theory for Normal Operators: A Complete Set of Unitary Invariants
CHAPTER X Unbounded Operators
1. Basic Properties and Examples
2. Symmetric and Self-Adjoint Operators
3. The Cayley Transform
4. Unbounded Normal Operators and the Spectral Theorem
5. Stone's Theorem
6. The Fourier Transform and Differentiation
7. Moments
CHAPTER XI Fredholm Theory
1. The Spectrum Revisited
2. Fredholm Operators
3. The Fredholm Index
4. The Essential Spectrum
5. The Components of SF
6. A Finer Analysis of the Spectrum
APPENDIX A. Preliminaries
1. Linear Algebra
2. Topology
APPENDIX B. The Dual of Lp(μ)
APPENDIX C. The Dual of C0(X)
Bibliography
List of Symbols
Index