Functional Analysis

Description

This second edition is thoroughly revised and includes several new examples and exercises. Proofs of many results have been rewritten for a greater clarity. While covering all the standard material expected of such a course, efforts have been made to illustrate the use of the topics to study differential equations and calculus of variations. The book includes a chapter on weak topologies and their applications. It also includes a chapter on the Lebesgue spaces, which discusses Sobolev spaces. The book includes a chapter on compact operators and their spectra, especially for compact self-adjoint operators on a Hilbert space. Each chapter has a large collection of exercises in the end, which give additional examples and counterexamples to the results given in the text. This book is suitable for a first course in functional analysis for graduate students who wish to pursue a career in the applications of mathematics. 1 Preface Preface to the Second Edition Notations Contents About the Author 978-981-19-7633-9_1 1 Preliminaries 1.1 Linear Spaces 1.2 Topological Spaces 1.3 Measure and Integration References 978-981-19-7633-9_2 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 978-981-19-7633-9_3 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 978-981-19-7633-9_4 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 978-981-19-7633-9_5 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 978-981-19-7633-9_6 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 978-981-19-7633-9_7 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 978-981-19-7633-9_8 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 1 (1) Index