Introduction to the Analysis of Metric Spaces

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Paperback | 2 November 1987

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Assuming a basic knowledge of real analysis and linear algebra, the student is given some familiarity with the axiomatic method in analysis and is shown the power of this method in exploiting the fundamental analysis structures underlying a variety of applications. Although the text is titled metric spaces, normed linear spaces are introduced immediately because this added structure is present in many examples and its recognition brings an interesting link with linear algebra; finite dimensional spaces are discussed earlier. It is intended that metric spaces be studied in some detail before general topology is begun. This follows the teaching principle of proceeding from the concrete to the more abstract. Graded exercises are provided at the end of each section and in each set the earlier exercises are designed to assist in the detection of the abstract structural properties in concrete examples while the latter are more conceptually sophisticated.

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Preface	p. viii
Metric Spaces and Normed Linear Spaces	p. 1
Definitions and Examples	p. 1
Metric spaces, normed linear spaces
Metrics generated by a norm
Co-ordinate, sequence and function spaces
Semi-normed linear spaces
Exercises
Balls and Boundedness	p. 21
Balls and spheres in metric spaces and normed linear spaces, relating norms and balls
Boundedness, diameter
Distances between sets
Exercises
Limit Processes	p. 36
Convergence and Completeness	p. 36
Convergence of sequences, characterisation in finite dimensional normed linear spaces, uniform convergence
Equivalent metrics and norms
Cauchy sequences, completeness
Convergence of series
Exercises
Cluster Points and Closure	p. 66
Cluster points, closed sets
Relating closed to complete
Closure, density, separability
The boundary of a set
Exercises
Application: Banach's Fixed Point Theorem	p. 91
Fixed points, Banach's Fixed Point Theorem
Application in real analysis	p. 93
Application in linear algebra	p. 96
Application in the theory of differential equations	p. 100
Picard's Theorem
Application in the theory of integral equations	p. 103
Fredholm integral equations, Volterra integral equations
Exercises
Continuity	p. 114
Continuity in Metric Spaces	p. 114
Local continuity, characterisation of continuity by sequences, algebra of continuous mappings
Global continuity characterised by inverse images
Isometrics, homeomorphisms
Uniform continuity
Exercises
Continuous Linear Mappings	p. 138
Characterisation of continuity of linear mappings, linear mappings on finite dimensional normed linear spaces, continuity of linear functionals
Topological isomorphisms, isometric isomorphisms
Exercises
Compactness	p. 160
Sequential Compactness in Metric Spaces	p. 161
Properties of compact sets
Characterisation in finite dimensional normed linear spaces, Riesz Theorem
Application in approximation theory
Alternative forms of compactness, total boundedness, ball cover compactness
Separability
Exercises
Continuous Functions on Compact Metric Spaces	p. 183
Heine's Theorem, Dini's Theorem
The structure of the real Banach space (C [a, b], [double vertical bar][middle dot][double vertical bar][subscript infinity]	p. 187
The Weierstrass Approximation Theorem
The structure of the Banach space (C(X), [double vertical bar][middle dot][double vertical bar][subscript infinity] where (X, d) is a compact metric space	p. 194
Compactness in (C(X), [double vertical bar][middle dot][double vertical bar][subscript infinity]	p. 200
Equicontinuity, The Ascoli-Arzela Theorem, Peano's Theorem
Exercises
The Metric Topology	p. 213
The Topological Analysis of Metric Spaces	p. 214
Open sets and their properties, base for a topology
Equivalent metrics
Relation to closed sets
The interior of a set
The characterisation of continuous mappings by inverse images
Topological compactness
Separability, the normal topological structure
Exercises
Appendices	p. 235
The real analysis background	p. 235
The set theory background	p. 240
The linear algebra background	p. 246
Index to Notation	p. 251
Index	p. 253
Table of Contents provided by Syndetics. All Rights Reserved.

Introduction to the Analysis of Metric Spaces

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Paperback

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