This book constitutes a concise introductory course on Functional Analysis for students who have studied calculus and linear algebra. The topics covered are Banach spaces, continuous linear transformations, Frechet derivative, geometry of Hilbert spaces, compact operators, and distributions. In addition, the book includes selected applications of functional analysis to differential equations, optimization, physics (classical and quantum mechanics), and numerical analysis. The book contains 197 problems, meant to reinforce the fundamental concepts.
The inclusion of detailed solutions to all the exercises makes the book ideal also for self-study. A Friendly Approach to Functional Analysis is written specifically for undergraduate students of pure mathematics and engineering, and those studying joint programmes with mathematics.
Author(s): Amol Sasane
Series: Essential Textbooks in Mathematics
Publisher: World Scientific Publishing
Year: 2017
Language: English
Pages: 396
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 (ā) Open Mapping Theorem
2.6 Spectral Theory
2.7 (ā) 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 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 K(X, Y) of all compact operators
5.3 Approximation of compact operators
5.4 (ā) 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 solutions
6.4 Multiplication by Cā functions
6.5 Fourier transform of (tempered) distributions
Solutions
The Lebesgue integral
Bibliography
Index
