Bibliothèque de L'IMIST/CNRST

Description based on print version record.

Includes bibliographical references (p. 409-416) and index.

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 readers understanding with numerous end-of-chapter problems. 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.

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 -- 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 -- References -- List of Symbols -- Index.