Real Variables with Basic Metric Space Topology

by Ash, Robert B.

• ISBN13: 9780486472201
• ISBN10: 0486472205
• Format: Paperback
• Copyright: 2009-05-21
• Publisher: Dover Publications

Summary

Designed for a first course in real variables, this text presents the fundamentals for more advanced mathematical work, particularly in the areas of complex variables, measure theory, differential equations, functional analysis, and probability. Geared toward advanced undergraduate and graduate students of mathematics, it is also appropriate for students of engineering, physics, and economics who seek an understanding of real analysis.

Author Biography

A Professor Emeritus of Mathematics at the University of Illinois, Robert Ash is the author of three other Dover books: Basic Abstract Algebra, Basic Probability Theory, and Complex Variables and Information Theory.

Table of Contents

Preface	p. ix
Introduction	p. 1
Basic Terminology	p. 1
Finite and Infinite Sets; Countably Infinite and Uncountably Infinite Sets	p. 7
Distance and Convergence	p. 10
Minicourse in Basic Logic	p. 15
Limit Points and Closure	p. 21
Review Problems for Chapter 1	p. 23
Some Basic Topological Properties of Rp	p. 25
Unions and Intersections of Open and Closed Sets	p. 25
Compactness	p. 28
Some Applications of Compactness	p. 33
Least Upper Bounds and Completeness	p. 37
Review Problems for Chapter 2	p. 42
Upper and Lower Limits of Sequences of Real Numbers	p. 45
Generalization of the Limit Concept	p. 45
Some Properties of Upper and Lower Limits	p. 49
Convergence of Power Series	p. 52
Review Problems for Chapter 3	p. 56
Continuous Functions	p. 57
Continuity: Ideas, Basic Terminology, Properties	p. 57
Continuity and Compactness	p. 64
Types of Discontinuities	p. 70
The Cantor Set	p. 76
Review Problems for Chapter 4	p. 80
Differentiation	p. 81
The Derivative and Its Basic Properties	p. 81
Additional Properties of the Derivative; Some Applications of the Mean Value Theorem	p. 86
Review Problems for Chapter 5	p. 92
Riemann-Stieltjes Integration	p. 93
Definition of the Integral	p. 93
Properties of the Integral	p. 98
Functions of Bounded Variation	p. 106
Some Useful Integration Theorems	p. 111
Review Problems for Chapter 6	p. 115
Uniform Convergence and Applications	p. 117
Pointwise and Uniform Convergence	p. 117
Uniform Convergence and Limit Operations	p. 122
The Weierstrass M-test and Applications	p. 125
Equicontinuity and the Arzela-Ascoli Theorem	p. 130
The Weierstrass Approximation Theorem	p. 134
Review Problems for Chapter 7	p. 139
Further Topological Results	p. 141
The Extension Problem	p. 141
Baire Category Theorem	p. 144
Connectedness	p. 150
Semicontinuous Functions	p. 152
Review Problems for Chapter 8	p. 158
Epilogue	p. 161
Some Compactness Results	p. 161
Replacing Cantor's Nested Set Property	p. 164
The Real Numbers Revisited	p. 165
Solutions to Problems	p. 167
Index	p. 207
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