Introduction to Hilbert Spaces with Applications

Dedication

Preface to the Third Edition

Preface to the Second Edition

Preface to the First Edition

Chapter 1: Normed Vector Spaces

1.1 Introduction

1.2 Vector Spaces

1.3 Normed Spaces

1.4 Banach Spaces

1.5 Linear Mappings

1.6 Contraction Mappings and the Banach Fixed Point Theorem

1.7 Exercises

Chapter 2: The Lebesgue Integral

2.1 Introduction

2.2 Step Functions

2.3 Lebesgue Integrable Functions

Definition 2.3.1. (Lebesgue integrable function)

Lemma 2.3.2.

2.4 The Absolute Value of an Integrable Function

2.5 Series of Integrable Functions

2.6 Norm in L¹()

Definition 2.6.1. (L1-norm)

Definition 2.6.2. (Null function)

Theorem 2.6.3.

Definition 2.6.4. (Convergence in norm)

Theorem 2.6.5.

Theorem 2.6.6.

2.7 Convergence Almost Everywhere

2.8 Fundamental Convergence Theorems

2.9 Locally Integrable Functions

2.10 The Lebesgue Integral and the Riemann Integral

2.11 Lebesgue Measure on

2.12 Complex-Valued Lebesgue Integrable Functions

2.13 The Spaces L^p()

2.14 Lebesgue Integrable Functions on N

2.15 Convolution

2.16 Exercises

Chapter 3: Hilbert Spaces and Orthonormal Systems

3.1 Introduction

3.2 Inner Product Spaces

3.3 Hubert Spaces

3.4 Orthogonal and Orthonormal Systems

3.5 Trigonometric Fourier Series

3.6 Orthogonal Complements and Projections

3.7 Linear Functional and the Riesz Representation Theorem

3.8 Exercises

Chapter 4: Linear Operators on Hilbert Spaces

4.1 Introduction

4.2 Examples of Operators

4.3 Bilinear Functional and Quadratic Forms

4.4 Adjoint and Self-Adjoint Operators

4.5 Invertible, Normal, Isometric, and Unitary Operators

4.6 Positive Operators

4.7 Projection Operators

4.8 Compact Operators

4.9 Eigenvalues and Eigenvectors

4.10 Spectral Decomposition

4.11 Unbounded Operators

4.12 Exercises

Chapter 5: Applications to Integral and Differential Equations

5.1 Introduction

5.2 Basic Existence Theorems

5.3 Fredholm Integral Equations

5.4 Method of Successive Approximations

5.5 Volterra Integral Equations

5.6 Method of Solution for a Separable Kernel

5.7 Volterra Integral Equations of the First Kind and Abel’s Integral Equation

5.8 Ordinary Differential Equations and Differential Operators

5.9 Sturm-Liouville Systems

5.10 Inverse Differential Operators and Green’s Functions

5.11 The Fourier Transform

5.12 Applications of the Fourier Transform to Ordinary Differential Equations and Integral Equations

5.13 Exercises

Chapter 6: Generalized Functions and Partial Differential Equations

6.1 Introduction

6.2 Distributions

6.3 Sobolev Spaces

6.4 Fundamental Solutions and Green’s Functions for Partial Differential Equations

6.5 Weak Solutions of Elliptic Boundary Value Problems

6.6 Examples of Applications of the Fourier Transform to Partial Differential Equations

6.7 Exercises

Chapter 7: Mathematical Foundations of Quantum Mechanics

7.1 Introduction

7.2 Basic Concepts and Equations of Classical Mechanics

7.3 Basic Concepts and Postulates of Quantum Mechanics

7.4 The Heisenberg Uncertainty Principle

7.5 The Schrödinger Equation of Motion

7.6 The Schrödinger Picture

7.7 The Heisenberg Picture and the Heisenberg Equation of Motion

7.8 The Interaction Picture

7.9 The Linear Harmonic Oscillator

7.10 Angular Momentum Operators

7.11 The Dirac Relativistic Wave Equation

7.12 Exercises

Chapter 8: Wavelets and Wavelet Transforms

8.1 Brief Historical Remarks

8.2 Continuous Wavelet Transforms

8.3 The Discrete Wavelet Transform

8.4 Multiresolution Analysis and Orthonormal Bases of Wavelets

8.5 Examples of Orthonormal Wavelets

8.6 Exercises

Chapter 9: Optimization Problems and Other Miscellaneous Applications

9.1 Introduction

9.2 The Gateaux and Fréchet Differentials

9.3 Optimization Problems and the Euler-Lagrange Equations

9.4 Minimization of Quadratic Functional

9.5 Variational Inequalities

9.6 Optimal Control Problems for Dynamical Systems

9.7 Approximation Theory

9.8 The Shannon Sampling Theorem

9.9 Linear and Nonlinear Stability

9.10 Bifurcation Theory

9.11 Exercises

Hints and Answers to Selected Exercises

Bibliography

Index