Contents

Preface

1.Normed and Banach spaces

1.1Vector spaces

1.2Normed spaces

1.3Topology of normed spaces

1.4Sequences in a normed space; Banach spaces

1.5Compact sets

2.Continuous and linear maps

2.1Linear transformations

2.2Continuous maps

2.3The normed space CL(X, Y)

2.4Composition of continuous linear transformations

2.5(∗) Open Mapping Theorem

2.6Spectral Theory

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

3.Differentiation

3.1Definition of the derivative

3.2Fundamental theorems of optimisation

3.3Euler-Lagrange equation

3.4An excursion in Classical Mechanics

4.Geometry of inner product spaces

4.1Inner product spaces

4.2Orthogonality

4.3Best approximation

4.4Generalised Fourier series

4.5Riesz Representation Theorem

4.6Adjoints of bounded operators

4.7An excursion in Quantum Mechanics

5.Compact operators

5.1Compact operators

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

5.3Approximation of compact operators

5.4(∗) Spectral Theorem for Compact Operators

6.A glimpse of distribution theory

6.1Test functions, distributions, and examples

6.2Derivatives in the distributional sense

6.3Weak solutions

6.4Multiplication by C^∞ functions

6.5Fourier transform of (tempered) distributions

Solutions

The Lebesgue integral

Bibliography

Index