Computational Functional Analysis, second edition
This book is devoted to a brief survey of basic structures and methods of functional analysis used in computational mathematics and numerical analysis. It is an introductory text intended for students at the first year graduate level; despite its occasional lack of depth and precision, it provides a good opportunity to get acquainted with the rudiments of this powerful discipline. The main emphasis is on numerical methods for operator equations - in particular, on analysis of approximation error in various methods for obtaining approximate solutions to equations and system of equations.
The book could be loosely divided into two parts; the former concentrates on linear operator equations, the latter on nonlinear operator equations. After introducing the basic framework used in functional analysis (such as linear, topological, metric, Banach and Hilbert spaces), the authors move on to linear functionals and operators and several types of convergence in Banach spaces. Special chapters are devoted to reproducing kernel Hilbert spaces and order relations in function spaces.
The first part of the book finishes with basic elements of the Fredholm theory of compact operators on Hilbert spaces and with approximation methods for linear operator equations. The second part (concentrating on nonlinear equations) starts with interval methods for operator equations and basic fixed point problems. After introducing Fréchet derivatives in Banach spaces and its elementary properties, their applications in Newton's method and its variant in infinite dimensional spaces are presented. The last chapter is devoted to a particular example of a use of the theory in a “real-world” problem; the authors choose a hybrid method for a free boundary problem. The topics are mostly discussed without proofs but each chapter is accompanied by a series of exercises that are designed to help students to learn how to discover mathematics for themselves. Hence the book serves as a readable introduction to functional analytical tools involved in computation.