Applications of Functional Analysis and Operator Theory

1. Banach Spaces1.1 Introduction1.2 Vector Spaces1.3 Normed Vector Spaces1.4 Banach Spaces1.5 Hilbert SpaceProblems2. Lebesgue Integration and the Lp Spaces2.1 Introduction2.2 The Measure of a Set2.3 Measurable Functions2.4 Integration2.5 The Lp Spaces2.6 ApplicationsProblems3. Foundations of Linear Operator Theory3.1 Introduction3.2 The Basic Terminology of Operator Theory3.3 Some Algebraic Properties of Linear Operators3.4 Continuity and Boundedness3.5 Some Fundamental Properties of Bounded Operators3.6 First Results on the Solution of the Equation Lf=g3.7 Introduction to Spectral Theory3.8 Closed Operators and Differential EquationsProblems4. Introduction to Nonlinear Operators4.1 Introduction4.2 Preliminaries4.3 The Contraction Mapping Principle4.4 The Frechet Derivative4.5 Newton's Method for Nonlinear OperatorsProblems5. Compact Sets in Banach Spaces5.1 Introduction5.2 Definitions5.3 Some Consequences of Compactness5.4 Some Important Compact Sets of FunctionsProblems6. The Adjoint Operator6.1 Introduction6.2 The Dual of a Banach Space6.3 Weak Convergence6.4 Hilbert Space6.5 The Adjoint of a Bounded Linear Operator6.6 Bounded Self-adjoint Operators -- Spectral Theory6.7 The Adjoint of an Unbounded Linear Operator in Hilbert SpaceProblems7. Linear Compact Operators7.1 Introduction7.2 Examples of Compact Operators7.3 The Fredholm Alternative7.4 The Spectrum7.5 Compact Self-adjoint Operators7.6 The Numerical Solution of Linear Integral EquationsProblems8. Nonlinear Compact Operators and Monotonicity8.1 Introduction8.2 The Schauder Fixed Point Theorem8.3 Positive and Monotone Operators in Partially Ordered Banach SpacesProblems9. The Spectral Theorem9.1 Introduction9.2 Preliminaries9.3 Background to the Spectral Theorem9.4 The Spectral Theorem for Bounded Self-adjoint Operators9.5 The Spectrum and the Resolvent9.6 Unbounded Self-adjoint Operators9.7 The Solution of an Evolution EquationProblems10. Generalized Eigenfunction Expansions Associated with Ordinary Differential Equations10.1 Introduction10.2 Extensions of Symmetric Operators10.3 Formal Ordinary Differential Operators: Preliminaries10.4 Symmetric Operators Associated with Formal Ordinary Differential Operators10.5 The Construction of Self-adjoint Extensions10.6 Generalized Eigenfunction ExpansionsProblems11. Linear Elliptic Partial Differential Equations11.1 Introduction11.2 Notation11.3 Weak Derivatives and Sobolev Spaces11.4 The Generalized Dirichlet Problem11.5 Fredholm Alternative for Generalized Dirichlet Problem11.6 Smoothness of Weak Solutions11.7 Further DevelopmentsProblems12. The Finite Element Method12.1 Introduction12.2 The Ritz Method12.3 The Rate of Convergence of the Finite Element MethodProblems13. Introduction to Degree Theory13.1 Introduction13.2 The Degree in Finite Dimensions13.3 The Leray-Schauder Degree13.4 A Problem in Radiative TransferProblems14. Bifurcation Theory14.1 Introduction14.2 Local Bifurcation Theory14.3 Global Eigenfunction TheoryProblems