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.