Intro
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 CL(X, Y)
2.4 Composition of continuous linear transformations
2.5 (aฬ#x80
ร) Open Mapping Theorem
2.6 Spectral Theory
2.7 (aฬ#x80
ร) 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 Mechanics4. Geometry of inner product spaces
4.1 Inner product spaces
4.2 Orthogonality
4.3 Best approximation
4.4 Generalised Fourier 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 KpX, Y q of all compact operators
5.3 Approximation of compact operators
5.4 (aฬ#x80
ร) Spectral Theorem for 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 solutions6.4 Multiplication by Caฬ#x88
#x9E
functions
6.5 Fourier transform of (tempered) distributions
Solutions
The Lebesgue integral
Bibliography
Index.