# Applications of Functional Analysis and Operator Theory

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Functional analysis is a powerful tool when applied to mathematical problems arising from physical situations. The present book provides, by careful selection of material, a collection of concepts and techniques essential for the modern practitioner. Emphasis is placed on the solution of equations (including nonlinear and partial differential equations). The assumed background is limited to elementary real variable theory and finite-dimensional vector spaces.

## Key Features

- Provides an ideal transition between introductory math courses and advanced graduate study in applied mathematics, the physical sciences, or engineering
- Gives the reader a keen understanding of applied functional analysis, building progressively from simple background material to the deepest and most significant results
- Introduces each new topic with a clear, concise explanation
- Includes numerous examples linking fundamental principles with applications
- Solidifies the reader's understanding with numerous end-of-chapter problems

## Readership

Graduate and prost-graduate students, researchers, teachers and professors

## Table of Contents

- 1. Banach Spaces

1.1 Introduction

1.2 Vector Spaces

1.3 Normed Vector Spaces

1.4 Banach Spaces

1.5 Hilbert Space

Problems

2. Lebesgue Integration and the Lp Spaces

2.1 Introduction

2.2 The Measure of a Set

2.3 Measurable Functions

2.4 Integration

2.5 The Lp Spaces

2.6 Applications

Problems

3. Foundations of Linear Operator Theory

3.1 Introduction

3.2 The Basic Terminology of Operator Theory

3.3 Some Algebraic Properties of Linear Operators

3.4 Continuity and Boundedness

3.5 Some Fundamental Properties of Bounded Operators

3.6 First Results on the Solution of the Equation Lf=g

3.7 Introduction to Spectral Theory

3.8 Closed Operators and Differential Equations

Problems

4. Introduction to Nonlinear Operators

4.1 Introduction

4.2 Preliminaries

4.3 The Contraction Mapping Principle

4.4 The Frechet Derivative

4.5 Newton's Method for Nonlinear Operators

Problems

5. Compact Sets in Banach Spaces

5.1 Introduction

5.2 Definitions

5.3 Some Consequences of Compactness

5.4 Some Important Compact Sets of Functions

Problems

6. The Adjoint Operator

6.1 Introduction

6.2 The Dual of a Banach Space

6.3 Weak Convergence

6.4 Hilbert Space

6.5 The Adjoint of a Bounded Linear Operator

6.6 Bounded Self-adjoint Operators -- Spectral Theory

6.7 The Adjoint of an Unbounded Linear Operator in Hilbert Space

Problems

7. Linear Compact Operators

7.1 Introduction

7.2 Examples of Compact Operators

7.3 The Fredholm Alternative

7.4 The Spectrum

7.5 Compact Self-adjoint Operators

7.6 The Numerical Solution of Linear Integral Equations

Problems

8. Nonlinear Compact Operators and Monotonicity

8.1 Introduction

8.2 The Schauder Fixed Point Theorem

8.3 Positive and Monotone Operators in Partially Ordered Banach Spaces

Problems

9. The Spectral Theorem

9.1 Introduction

9.2 Preliminaries

9.3 Background to the Spectral Theorem

9.4 The Spectral Theorem for Bounded Self-adjoint Operators

9.5 The Spectrum and the Resolvent

9.6 Unbounded Self-adjoint Operators

9.7 The Solution of an Evolution Equation

Problems

10. Generalized Eigenfunction Expansions Associated with Ordinary Differential Equations

10.1 Introduction

10.2 Extensions of Symmetric Operators

10.3 Formal Ordinary Differential Operators: Preliminaries

10.4 Symmetric Operators Associated with Formal Ordinary Differential Operators

10.5 The Construction of Self-adjoint Extensions

10.6 Generalized Eigenfunction Expansions

Problems

11. Linear Elliptic Partial Differential Equations

11.1 Introduction

11.2 Notation

11.3 Weak Derivatives and Sobolev Spaces

11.4 The Generalized Dirichlet Problem

11.5 Fredholm Alternative for Generalized Dirichlet Problem

11.6 Smoothness of Weak Solutions

11.7 Further Developments

Problems

12. The Finite Element Method

12.1 Introduction

12.2 The Ritz Method

12.3 The Rate of Convergence of the Finite Element Method

Problems

13. Introduction to Degree Theory

13.1 Introduction

13.2 The Degree in Finite Dimensions

13.3 The Leray-Schauder Degree

13.4 A Problem in Radiative Transfer

Problems

14. Bifurcation Theory

14.1 Introduction

14.2 Local Bifurcation Theory

14.3 Global Eigenfunction Theory

Problems

## Product details

- No. of pages: 432
- Language: English
- Copyright: © Elsevier Science 2005
- Published: February 8, 2005
- Imprint: Elsevier Science
- eBook ISBN: 9780080527314

## About the Authors

### V. Hutson

#### Affiliations and Expertise

University of Sheffield, UK

### J. Pym

#### Affiliations and Expertise

University of Sheffield, UK

### M. Cloud

#### Affiliations and Expertise

Lawrence Technological University, Southfield, USA

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