A First Look at Numerical Functional Analysis

by Sawyer, W. W.

• ISBN13: 9780486478821
• ISBN10: 0486478823
• Format: Paperback
• Copyright: 2010-12-22
• Publisher: Dover Publications

Summary

Functional analysis arose from traditional topics of calculus and integral and differential equations. This accessible text by an internationally renowned teacher and author starts with problems in numerical analysis and shows how they lead naturally to the concepts of functional analysis.

Author Biography

Internationally renowned for his innovative teaching methods, the late W. W. Sawyer was a Professor of Mathematics and Education at the University of Toronto. Dover has reprinted Sawyer's The Mathematician's Delight, which has been in print continuously since 1943 and has sold over 500,000 copies.

Table of Contents

List of Symbols	p. ix
A First Course in Functional Analysis	p. 1
'Soft' and 'hard' analysis	p. 3
Old Ideas in New Contexts	p. 5
Examples of iteration	p. 5
Functions, ancient and modern	p. 7
Continuity and distance	p. 11
Limits and convergence	p. 17
The role of the distance axioms	p. 24
Vectors	p. 25
Pure vector theory	p. 27
Axioms of a vector space	p. 28
Iteration and Contraction Mappings	p. 31
Testing convergence	p. 31
The idea of a contraction mapping	p. 33
The space of continuous functions	p. 38
Iteration and integral equations	p. 46
A matrix iteration	p. 50
Minkowski Spaces	p. 53
Introduction	p. 53
Analysis of an obscure proof	p. 54
A general construction for a norm	p. 58
Linear Operators and Their Norms	p. 62
Linear operators	p. 62
Operator norms	p. 65
The space of bounded linear operators	p. 70
Some operator norms	p. 74
Properties of operators	p. 77
Applications of operator norms	p. 79
Continuity of bounded operators	p. 86
Differentiation and Integration	p. 87
A classical theory of iteration	p. 87
Differentiation generalized	p. 88
Generalizing the mean value theorem	p. 92
Some worked examples	p. 98
The idea of an integral	p. 103
The Newton-Raphson method	p. 108
Further Developments	p. 120
Taylor series	p. 120
Differentiation and linear functions	p. 122
Majorants	p. 123
Polynomial operators	p. 125
Quadratic majorants	p. 127
Euclidean Space	p. 131
On perpendicularity	p. 131
Space of n dimensions	p. 134
Space of infinite dimensions	p. 138
Fourier series and orthogonal functions	p. 141
The theoretical background	p. 148
Applications of Hilbert space	p. 157
Some Tools of the Trade	p. 159
Geometry and generalization	p. 159
Calculus and inequalities	p. 162
Continuity, distance, norm, scalar product	p. 165
Some Bridgeheads	p. 167
Functional and the dual space	p. 167
An existence theorem	p. 170
An application of functionals	p. 171
Compactness	p. 173
Vector products	p. 176
Bibliography	p. 179
Index	p. 181
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