Book:Amol Sasane/A Friendly Approach to Functional Analysis

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Amol Sasane: A Friendly Approach to Functional Analysis

Published $2017$, World Scientific Publishing

ISBN 978-1786343338.


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