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Publisher:

W. H. Freeman

Publication Date:

1993

Number of Pages:

640

Format:

Hardcover

Edition:

2

Price:

159.95

ISBN:

978-0-7167-2105-5

Category:

Textbook

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by , on ]

Allen Stenger

11/25/2012

This is an introductory text in real analysis, aimed at upper-division undergraduates. The coverage is similar to that in Rudin’s Principles of Mathematical Analysis and Apostol’s Mathematical Analysis. This book differs from these earlier books primarily in being more talkative: explanations are written out at greater length, there are more worked examples, and there is a much larger number of exercises at all levels of difficulty.

One unusual feature of the present book is that in the body of each chapter the theorems are motivated, stated in detail, and illustrated through examples — but not proven. Complete proofs are given in a lengthy appendix to each chapter. The chapters are like survey articles, with supplements. I wasn’t completely satisfied with this approach, because at this level of course we are primarily interested in the proofs, but it is workable.

Some topics are omitted. Functions of several variables are included, but not vector calculus (vector-valued functions). The integration is strictly Riemannian, although sets of measure zero are included, as are limited versions of the Lebesgue convergence theorems.

Apart from the segregation of the proofs, I had just a couple of minor gripes. The book uses the reversed bracket notation for open intervals, for example ]0, 1[ is the open interval from 0 to 1. It also uses a mixed reference style, where references cited in the body have their bibliographic information given there, while there is a separate, disjoint list of References and Suggestions for Further Study.

Although the material is classical, it is not what most people call “classical analysis,”. The book does not deal with classical topics such as inequalities, approximation of functions, and special numbers and functions (as does, for example, Duren’s recent Invitation to Classical Analysis).

Bottom line: A good choice for students who find Rudin too austere.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.

Supplement on the Axioms of Set Theory

Ordered Fields and the Number Systems

Completeness and the Real Number System

Least Upper Bounds

Cauchy Sequences

Cluster Points: lim inf and lim sup

Euclidean Space

Norms, Inner Products, and Metrics

The Complex Numbers

Open Sets

Interior of a Set

Closed Sets

Accumulation Points

Closure of a Set

Boundary of a Set

Sequences

Completeness

Series of Real Numbers and Vectors

Compacted-ness

The Heine-Borel Theorem

Nested Set Property

Path-Connected Sets

Connected Sets

Continuity

Images of Compact and Connected Sets

Operations on Continuous Mappings

The Boundedness of Continuous Functions of Compact Sets

The Intermediate Value Theorem

Uniform Continuity

Differentiation of Functions of One Variable

Integration of Functions of One Variable

Pointwise and Uniform Convergence

The Weierstrass M Test

Integration and Differentiation of Series

The Elementary Functions

The Space of Continuous Functions

The Arzela-Ascoli Theorem

The Contraction Mapping Principle and Its Applications

The Stone-Weierstrass Theorem

The Dirichlet and Abel Tests

Power Series and Cesaro and Abel Summability

Definition of the Derivative

Matrix Representation

Continuity of Differentiable Mappings; Differentiable Paths

Conditions for Differentiability

The Chain Rule

Product Rule and Gradients

The Mean Value Theorem

Taylor's Theorem and Higher Derivatives

Maxima and Minima

Inverse Function Theorem

Implicit Function Theorem

The Domain-Straightening Theorem

Further Consequences of the

Implicit Function Theorem

An Existence Theorem for Ordinary Differential Equations

The Morse Lemma

Constrained Extrema and Lagrange Multipliers

Integrable Functions

Volume and Sets of Measure Zero

Lebesgue's Theorem

Properties of the Integral

Improper Integrals

Some Convergence Theorems

Introduction to Distributions

Introduction

Fubini's Theorem

Change of Variables Theorem

Polar Coordinates

Spherical Coordinates and Cylindrical Coordinates

A Note on the Lebesgue Integral

Interchange of Limiting Operations

Inner Product Spaces

Orthogonal Families of Functions

Completeness and Convergence Theorems

Functions of Bounded Variation and Fejér Theory (Optional)

Computation of Fourier Series

Further Convergence Theorems

Applications

Fourier Integrals

Quantum Mechanical Formalism

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