# Applications of Functional Analysis and Operator Theory

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 readerâs understanding with numerous end-of-chapter problems.

Audience

Hardbound, 432 Pages

Published: February 2005

Imprint: Elsevier

ISBN: 978-0-444-51790-6

## Contents

• Preface.Acknowledgements.Contents.1. Banach Spaces 1.1 Introduction 1.2 Vector Spaces 1.3 Normed Vector Spaces 1.4 Banach Spaces 1.5 Hilbert Space Problems2. 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 Problems3. 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 Problems4. 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 Problems5. 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 Problems6. 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 Problems7. 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 Problems8. 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 Problems9. 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 Problems10. 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 Problems11. 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 Problems12. The Finite Element Method 12.1 Introduction 12.2 The Ritz Method 12.3 The Rate of Convergence of the Finite Element Method Problems13. 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 Problems14. Bifurcation Theory 14.1 Introduction 14.2 Local Bifurcation Theory 14.3 Global Eigenfunction Theory Problems References List of Symbols Index