Applications of Functional Analysis and Operator Theory, Volume 200

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

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.

• 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