Preface.
PART ONE: GENERAL THEORY
1. Topological Vector Space
Introduction
Separation properties
Linear Mappings
Finite-dimensional spaces
Metrization
Boundedness and continuity
Seminorms and local convexity
Quotient spaces
Examples
Exercises
2. Completeness
Baire category
The Banach-Steinhaus theorem
The open mapping theorem
The closed graph theorem
Bilinear mappings
Exercises
3. Convexity
The Hahn-Banach theorems
Weak topologies
Compact convex sets
Vector-valued integration
Holomorphic functions
Exercises
4. Duality in Banach Spaces
The normed dual of a normed space
Adjoints
Compact operators
Exercises
5. Some Applications
A continuity theorem
Closed subspaces of Lp-spaces
The range of a vector-valued measure
A generalized Stone-Weierstrass theorem
Two interpolation theorems
Kakutani's fixed point theorem
Haar measure on compact groups
Uncomplemented subspaces
Sums of Poisson kernels
Two more fixed point theorems
Exercises
PART TWO: DISTRIBUTIONS AND FOURIER TRANSFORMS
6. Test Functions and Distributions
Introduction
Test function spaces
Calculus with distributions
Localization
Supports of distributions
Distributions as derivatives
Convolutions
Exercises
7. Fourier Transforms
Basic properties
Tempered distributions
Paley-Wiener theorems
Sobolev's lemma
Exercises
8. Applications to Differential Equations
Fundamental solutions
Elliptic equations
Exercises
9. Tauberian Theory
Wiener's theorem
The prime number theorem
The renewal equation
Exercises
PART THREE: BANACH ALGEBRAS AND SPECTRAL THEORY
10. Banach Algebras
Introduction
Complex homomorphisms
Basic properties of spectra
Symbolic calculus
The group of invertible elements
Lomonosov's invariant subspace theorem
Exercises
11. Commutative Banach Algebras
Ideals and homomorphisms
Gelfand transforms
Involutions
Applications to noncommutative algebras
Positive functionals
Exercises
12. Bounded Operators on a Hillbert Space
Basic facts
Bounded operators
A commutativity theorem
Resolutions of the identity
The spectral theorem
Eigenvalues of normal operators
Positive operators and square roots
The group of invertible operators
A characterization of B*-algebras
An ergodic theorem
Exercises
13. Unbounded Operators
Introduction
Graphs and symmetric operators
The Cayley transform
Resolutions of the identity
The spectral theorem
Semigroups of operators
Exercises
Appendix A: Compactness and Continuity
Appendix B: Notes and Comments
Bibliography
List of Special Symbols
Index
