Real Variables with Basic Metric Space Topology

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  • Format: Paperback
  • Copyright: 2009-05-21
  • Publisher: Dover Publications
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Supplemental Materials

What is included with this book?


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

Prefacep. ix
Introductionp. 1
Basic Terminologyp. 1
Finite and Infinite Sets; Countably Infinite and Uncountably Infinite Setsp. 7
Distance and Convergencep. 10
Minicourse in Basic Logicp. 15
Limit Points and Closurep. 21
Review Problems for Chapter 1p. 23
Some Basic Topological Properties of Rpp. 25
Unions and Intersections of Open and Closed Setsp. 25
Compactnessp. 28
Some Applications of Compactnessp. 33
Least Upper Bounds and Completenessp. 37
Review Problems for Chapter 2p. 42
Upper and Lower Limits of Sequences of Real Numbersp. 45
Generalization of the Limit Conceptp. 45
Some Properties of Upper and Lower Limitsp. 49
Convergence of Power Seriesp. 52
Review Problems for Chapter 3p. 56
Continuous Functionsp. 57
Continuity: Ideas, Basic Terminology, Propertiesp. 57
Continuity and Compactnessp. 64
Types of Discontinuitiesp. 70
The Cantor Setp. 76
Review Problems for Chapter 4p. 80
Differentiationp. 81
The Derivative and Its Basic Propertiesp. 81
Additional Properties of the Derivative; Some Applications of the Mean Value Theoremp. 86
Review Problems for Chapter 5p. 92
Riemann-Stieltjes Integrationp. 93
Definition of the Integralp. 93
Properties of the Integralp. 98
Functions of Bounded Variationp. 106
Some Useful Integration Theoremsp. 111
Review Problems for Chapter 6p. 115
Uniform Convergence and Applicationsp. 117
Pointwise and Uniform Convergencep. 117
Uniform Convergence and Limit Operationsp. 122
The Weierstrass M-test and Applicationsp. 125
Equicontinuity and the Arzela-Ascoli Theoremp. 130
The Weierstrass Approximation Theoremp. 134
Review Problems for Chapter 7p. 139
Further Topological Resultsp. 141
The Extension Problemp. 141
Baire Category Theoremp. 144
Connectednessp. 150
Semicontinuous Functionsp. 152
Review Problems for Chapter 8p. 158
Epiloguep. 161
Some Compactness Resultsp. 161
Replacing Cantor's Nested Set Propertyp. 164
The Real Numbers Revisitedp. 165
Solutions to Problemsp. 167
Indexp. 207
Table of Contents provided by Ingram. All Rights Reserved.

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