# Introduction to Hilbert Spaces with Applications

## 3rd Edition

**Authors:**Lokenath Debnath Piotr Mikusinski

**eBook ISBN:**9780080455921

**Hardcover ISBN:**9780122084386

**Imprint:**Academic Press

**Published Date:**29th September 2005

**Page Count:**600

## Description

Building on the success of the two previous editions, *Introduction to Hilbert Spaces with Applications, Third Edition*, offers an overview of the basic ideas and results of Hilbert space theory and functional analysis. It acquaints students with the Lebesgue integral, and includes an enhanced presentation of results and proofs. Students and researchers will benefit from the wealth of revised examples in new, diverse applications as they apply to optimization, variational and control problems, and problems in approximation theory, nonlinear instability, and bifurcation. The text also includes a popular chapter on wavelets that has been completely updated. Students and researchers agree that this is the definitive text on Hilbert Space theory.

## Key Features

- Updated chapter on wavelets
- Improved presentation on results and proof
- Revised examples and updated applications
- Completely updated list of references

## Readership

2-semester course on Functional Analysis or Hilbert space course for junior-senior-grad math students, Also researchers and others interested in math theory.

## Table of Contents

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 *L*1(*)*

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 *Lp*(*)*

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

## Details

- No. of pages:
- 600

- Language:
- English

- Copyright:
- © Academic Press 2006

- Published:
- 29th September 2005

- Imprint:
- Academic Press

- eBook ISBN:
- 9780080455921

- Hardcover ISBN:
- 9780122084386

## About the Author

### Lokenath Debnath

Lokenath Debnath is Professor of the Department of Mathematics and Professor of Mechanical and Aerospace Engineering at the University of Central Florida in Orlando. He received his M.Sc. and Ph.D. degrees in pure mathematics from the University of Calcutta, and obtained D.I.C. and Ph.D. degrees in applied mathematics from the Imperial College of Science and Technology, University of London. He was a Senior Research Fellow at the University of Cambridge and has had visiting appointments to several universities in the United States and abroad. His many honors and awards include two Senior Fulbright Fellowships and an NSF Scientist award to visit India for lectures and research. Dr. Debnath is author or co-author of several books and research papers in pure and applied mathematics, and serves on several editorial boards for scientific journals. He is the current and founding Managing Editor of the *International Journal of Mathematics and Mathematical Sciences*.

### Affiliations and Expertise

University of Central Florida, Orlando, U.S.A.

### Piotr Mikusinski

Piotr Mikusinski received his Ph.D. in mathematics from the Institute of Mathematics of the Polish Academy of Sciences. In 1983, he became visiting lecturer at the University of California at Santa Barbara, where he spent two years. He is currently a member of the faculty in the Department of Mathematics at the University of Central Florida in Orlando. His main research interests are the theory of generalized functions and real analysis. He has published many research articles and is co-author with his father, Jan Mikusinski, of *An Introduction to Analysis: From Number to Integral*.

### Affiliations and Expertise

University of Central Florida, Orlando, U.S.A.

## Reviews

"...this is a very useful and good book and it can find a place in the library of anybody interested in functional analysis, particularly Hilbert Spaces and their applications." --*MAA REVIEWS*