Book:Srinivasan Kesavan/Functional Analysis

Subject Matter

 * Functional Analysis

Contents

 * 1. Preliminaries


 * 1.1 Linear Spaces


 * 1.2 Topological Spaces


 * 1.3 Measure and Integration


 * 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 Geometric Versions


 * 3.3 Vector Valued Integration


 * 3.4 An Application to Optimization Theory


 * 3.5 Exercises


 * 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


 * 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. $L^p$ Spaces


 * 6.1 Basic properties


 * 6.2 Duals of $L^p$ Spaces


 * 6.3 The Spaces $\map {L^p} \Omega$


 * 6.4 The Spaces $\map {W^{1,p} } {a,b}$


 * 6.5 Exercises


 * 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


 * 8. Compact Operators


 * 8.1 Basic Properties


 * 8.2 Riesz-Fredhölm Theory


 * 8.3 Spectrum of a Compact Operator


 * 8.4 Compact Self-Adjoint Operators


 * 8.5 Exercises

Bibliography

Index