A Friendly Approach To Functional Analysis

A Friendly Approach To Functional Analysis

by Amol Sasane

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Product Details

ISBN-13: 9781786343345
Publisher: World Scientific Publishing Europe Ltd
Publication date: 02/21/2017
Series: Essential Textbooks In Mathematics Series
Pages: 396
Sales rank: 1,085,111
Product dimensions: 5.90(w) x 8.90(h) x 1.00(d)

Table of Contents

Preface vii

1 Normed and Banach spaces 1

1.1 Vector spaces 3

1.2 Normed spaces 7

1.3 Topology of normed spaces 17

1.4 Sequences in a normed space; Banach spaces 24

1.5 Compact sets 44

2 Continuous and linear maps 53

2.1 Linear transformations 54

2.2 Continuous maps 58

2.3 The normed space CL(X,Y) 67

2.4 Composition of continuous linear transformations 82

2.5 (*) Open Mapping Theorem 92

2.6 Spectral Theory 97

2.7 (*) Dual space and the Hahn-Banach Theorem 104

3 Differentiation 117

3.1 Definition of the derivative 118

3.2 Fundamental theorems of optimisation 125

3.3 Euler-Lagrange equation 134

3.4 An excursion in Classical Mechanics 145

4 Geometry of inner product spaces 155

4.1 Inner product spaces 156

4.2 Orthogonality 165

4.3 Best approximation 174

4.4 Generalised Fourier series 183

4.5 Riesz Representation Theorem 189

4.6 Adjoints of bounded operators 190

4.7 An excursion in Quantum Mechanics 200

5 Compact operators 209

5.1 Compact, operators 210

5.2 The set K(X,Y) of all compact operators 211

5.3 Approximation of compact operators 217

5.4 (*) Spectral Theorem for Compact Operators 222

6 A glimpse of distribution theory 227

6.1 Test functions, distributions, and examples 230

6.2 Derivatives in the distributional sense 237

6.3 Weak solutions 243

6.4 Multiplication by C functions 248

6.5 Fourier transform of (tempered) distributions 253

Solutions 259

The Lebesgue integral 359

Bibliography 373

Index 375

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