Real Variables with Basic Metric Space Topology

Real Variables with Basic Metric Space Topology

by Robert B. Ash

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Overview


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.
The author encourages an intuitive approach to problem solving and offers concrete examples, diagrams, and geometric or physical interpretations of results. Detailed solutions to the problems appear within the text, making this volume ideal for independent study. Topics include metric spaces, Euclidean spaces and their basic topological properties, sequences and series of real numbers, continuous functions, differentiation, Riemann-Stieltjes integration, and uniform convergence and applications.

Product Details

ISBN-13: 9780486472201
Publisher: Dover Publications
Publication date: 05/21/2009
Series: Dover Books on Mathematics Series
Pages: 224
Product dimensions: 6.00(w) x 9.10(h) x 0.50(d)
Age Range: 18 Years

About the Author


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


INTRODUCTION
Basic Terminology
Finite and Infinite Sets; Countably Infinite and Uncountably Infinite Sets
Distance and Convergence
Minicourse in Basic Logic
Limit Points and Closure
Review Problems for Chapter 1
SOME BASIC TOPOLOGICAL PROPERTIES OF Rp
Unions and Intersections of Open and Closed Sets
Compactness
Some Applications of Compactness
Least Upper Bounds and Completeness
Review Problems for Chapter 2
UPPER AND LOWER LIMITS OF SEQUENCES OF REAL NUMBERS
Generalization of the Limit Concept
Some Properties of Upper and Lower Limits
Convergence of Power Series
Review Problems for Chapter 3
CONTINUOUS FUNCTIONS
Continuity: Ideas, Basic Terminology, Properties
Continuity and Compactness
Types of Discontinuities
The Cantor Set
Review Problems for Chapter 4
DIFFERENTIATION
The Derivative and Its Basic Properties
Additional Properties of the Derivative; Some Applications of the Mean Value Theorem
Review Problems for Chapter 5
RIEMANN-STIELTJES INTEGRATION
Definition of the Integral
Properties of the Integral
Functions of Bounded Variation
Some Useful Integration Theorems
Review Problems for Chapter 6
UNIFORM CONVERGENCE AND APPLICATIONS
Pointwise and Uniform Convergence
Uniform Convergence and Limit Operations
The Weierstrass M-test and Applications
Equicontinuity and the Arzela-Ascoli Theorem
The Weierstrass Approximation Theorem
Review Problems for Chapter 7
FURTHER TOPOLOGICAL RESULTS
The Extension Problem
Baire Category Theorem
Connectedness
Semicontinuous Functions
Review Problems for Chapter 8
EPILOGUE
Some Compactness Results
Replacing Cantor's Nested Set Property
The Real Numbers Revisited
SOLUTIONS TO PROBLEMS
INDEX

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