Book:Amol Sasane/A Friendly Approach to Functional Analysis

Subject Matter

 * Functional Analysis

Contents
Preface

1. Normed and Banach spaces


 * 1.1 Vector spaces


 * 1.2 Normed spaces


 * 1.3 Topology of normed spaces


 * 1.4 Sequences in a normed space; Banach spaces


 * 1.5 Compact sets

2. Continuous and linear maps


 * 2.1 Linear transformations


 * 2.2 Continuous maps


 * 2.3 The normed space $\map {CL} {X,Y}$


 * 2.4 Composition of continuous linear transformations


 * 2.5 (*) Open Mapping Theorem


 * 2.6 Spectral Theory


 * 2.7 (*) Dual space and the Hahn-Banach Theorem

3. Differentiation


 * 3.1 Definition of the derivative


 * 3.2 Fundamental theorems of optimisation


 * 3.3 Euler-Lagrange equation


 * 3.4 An excursion in Classical Mechanics

4. Geometry of inner product spaces


 * 4.1 Inner product spaces


 * 4.2 Orthogonality


 * 4.3 Best approximation


 * 4.4 Generalised Fourie series


 * 4.5 Riesz Representation Theorem


 * 4.6 Adjoints of bounded operators


 * 4.7 An excursion in Quantum Mechanics

5. Compact operators


 * 5.1 Compact operators


 * 5.2 The set $\map K {X,Y}$ of all compact operators


 * 5.3 Approximation of compact operators


 * 5.4 (*) Spectral Theorem of Compact Operators

6. A glimpse of distribution theory


 * 6.1 Test functions, distributions, and examples


 * 6.2 Derivatives in the distributional sense


 * 6.3 Weak solutions


 * 6.4 Multiplication by $C^\infty$ functions


 * 6.5 Fourier transform of (tempered) distributions

Solutions

The Lebesgue integral

Bibliography

Index