Book:Srinivasan Kesavan/Functional Analysis

Srinivasan Kesavan: Functional Analysis

Published $\text {2009}$, Hindustan Book Agency

ISBN 978-8185931876

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