Applications of Functional Analysis and Operator Theory

Applications of Functional Analysis and Operator Theory

2nd Edition - February 8, 2005

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  • Authors: V. Hutson, J. Pym, M. Cloud
  • eBook ISBN: 9780080527314

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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


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

    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

    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

    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

    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

    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

    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

    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

    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

    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

    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

    12. The Finite Element Method

    12.1 Introduction
    12.2 The Ritz Method
    12.3 The Rate of Convergence of the Finite Element Method

    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

    14. Bifurcation Theory

    14.1 Introduction
    14.2 Local Bifurcation Theory
    14.3 Global Eigenfunction Theory

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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