This book is the first of a multivolume series devoted to an exposition of functional analysis methods in modem mathematical physics.It describes the fundamental principles of functional analysis and is essentially self-contained,although there are occasional references to later volumes.We have included a few applications when we thought that they would provide motivation for the reader.Later volumes describe various advanced topics in functional analysis and give numerous applications in classical physics,modem physics,and partial differenrial equations.

Preface

Introduction

Contents of Other Volumes

Ⅰ: PRELIMINARIES

1. Sets and functions

2. Metric and normed linear spaces

Appendix Lira sup and lim inf

3. The Lebesgue integral

4. Abstract measure theory

5. Two conrergence arguments

6. Equicontinuity

Notes

Problems

Ⅱ: HILBERT SPACES

1. The geometry of Hilbert space

2. The Riesz lemma

3. Orthonormal bases

4. Tensor products of Hilbert spaces

5. Ergodic theory: an introduction

Notes

Problems

Ⅲ: BANACH SPACES

1. Definition and examples

2. Duals and double duals

3. The Hahn-Banach theorem

4. Operations on Banach spaces

5. The Baire category theorem and its consequences

Notes

Problems

Ⅳ: TOPOLOGICAL SPACES

1. General notions

2. Nets and convergence

3. Compactness

Appendix The Stone-Weierstrass theorem

4. Measure theory on compact spaces

5. Weak topologies on Banach spaces

Appendix Weak and strong measurability

Notes

Problems

Ⅴ: LOCALLY CONVEX SPACES

1. General properties

2. Frdchet spaces

3. Functions of rapid decease and the tempered distributions

Appendix The N-representation for and

4. Inductive limits: generalized functions and weak solutions of partial differential equations

5. Fixed point theorems

6. Applications of fixed point theorems

7. Topologies on locally convex spaces: duality theory and the strong dual topology

Appendix Polars and the Mackey-Arens theorem

Notes

Problems

Ⅵ: BOUNDED OPERATORS

1. Topologies on bounded operators

2. Adjoints

3. The spectrum

4. Positive operators and the polar decomposition

5. Compact operators

6. The trace class and Hilbert-Schmidt ideals

Notes

Problems

Ⅶ: THE SPECTRAL THEOREM

1. The continuous functional calculus

2. The spectral measures

3. Spectral projections

4. Ergodic theory revisited: Koopmanism

Notes

Problems

Ⅷ: UNBOUNDED OPERATORS

1. Domains,graphs,adjoints,and spectrum

2. Symmetric and self-adjoint operators: the basic criterion for self-adjointness

3. The spectral theorem

4. Stone's theorem

5. Formal manipulation is a touchy business: Nelson's example

6. Quadratic forms

7. Convergence of unbounded operators

8. The Trotter product formula

9. The polar decomposition for closed operators

10. Tensor products

11. Three mathematical problems in quantum mechanics

Notes

Problems

THE FOURIER TRANSFORM

1. The Fourier transform on (Rn) and (Rn),convolutions

2. The range of the Fourier transform: Classical spaces

3. The range of the Fourier transform: Analyticity

Notes

Problems

SUPPLEMENTARY MATERIAL

Ⅱ.2. Applications of the Riesz lemma

Ⅲ.1. Basic properties of Lp spaces

Ⅳ.3. Proof of Tychonoff s theorem

Ⅳ.4. The Riesz-Markov theoremJor X = [0,1]

Ⅳ.5. Minimization of Junctionals

Ⅴ.5. Proofs of some theorems in nonlinear functional analysis

Ⅵ.5. Applications of compact operators

Ⅷ.7. Monotone convergence for forms

Ⅷ.8. More on the Trotter product formula Uses of the maximum principle

Notes

Problems

List of Symbols

Index