Functional Analysis

Description

This book is suited for a first course on Functional Analysis at the masters level. Efforts have been made to illustrate the use of various results via examples taken from differential equations and the calculus of variations, either through brief sections or through exercises. So, this book will be particularly useful for students who aspire to a research career in the applications of mathematics. Special emphasis has been given to the treatment of weak topologies and their applications to notions like reflexivity, separability and uniform convexity. The chapter on Lebesgue spaces includes a section devoted to the simplest examples of Sobolev spaces. The chapter on compact operators includes the spectral theory of compact self-adjoint operators on a Hilbert space. Each chapter concludes with a large selection of exercises of varying degrees of difficulty. They often provide examples or counter-examples to illustrate the optimality of the hypotheses of various theorems proved in the text, or develop simple versions of theories not developed therein. In this (second) edition, the book has been completely overhauled, without altering its original structure. Proofs of many results have been rewritten for greater clarity of exposition. Many examples have been added to make the text more user-friendly. Several new exercises have been added. Preface Preface to the Second Edition Notations Contents About the Author 1 Preliminaries 1.1 Linear Spaces 1.2 Topological Spaces 1.3 Measure and Integration References 2 Normed Linear Spaces 2.1 The Norm Topology 2.2 Examples 2.3 Continuous Linear Transformations 2.4 Applications to Differential Equations 2.5 Exercises 3 Hahn-Banach Theorems 3.1 Analytic Versions 3.2 Reflexivity 3.3 Geometric Versions 3.4 Vector-Valued Integration 3.5 An Application to Optimization Theory 3.6 Exercises Reference 4 Baire's Theorem and Applications 4.1 Baire's Theorem 4.2 Principle of Uniform Boundedness 4.3 Application to Fourier Series 4.4 The Open Mapping and Closed Graph Theorems 4.5 Annihilators 4.6 Complemented Subspaces 4.7 Unbounded Operators, Adjoints 4.8 Exercises Reference 5 Weak and Weak* Topologies 5.1 The Weak Topology 5.2 The Weak* Topology 5.3 Reflexive Spaces 5.4 Separable Spaces 5.5 Uniformly Convex Spaces 5.6 Application: Calculus of Variations 5.7 Exercises 6 Lp Spaces 6.1 Basic Properties 6.2 Duals of Lp Spaces 6.3 The Spaces Lp(Ω) 6.4 The Spaces W1,p(a,b) 6.5 Exercises References 7 Hilbert Spaces 7.1 Basic Properties 7.2 The Dual of a Hilbert Space 7.3 Application: Variational Inequalities 7.4 Orthonormal Sets 7.5 Exercises References 8 Compact Operators 8.1 Basic Properties 8.2 Riesz-Fredhölm Theory 8.3 Spectrum of an Operator 8.4 Spectrum of a Compact Operator 8.5 Compact Self-adjoint Operators 8.6 Exercises References Index