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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