# An Introduction to Analysis

Publisher:
Prentice Hall
Publication Date:
2010
Number of Pages:
680
Format:
Hardcover
Edition:
4
Price:
132.00
ISBN:
9780132296380
Category:
Textbook
[Reviewed by
Miklós Bóna
, on
06/10/2011
]

The book is meant to be a two-semester or three-quarter textbook for an Advanced Calculus course taken by last-year undergraduates or beginning graduate students who are not quite ready for a full graduate level course.

For a textbook with these goals, this book is on the short side. The proofs are rigorous, but brief. This is a very good approach as it will not scare the audience away. The number of exercises should be higher, however. There are usually at most ten of ten per section, often less. This is sufficient only for the good students who understand the concepts at their first try. More than half of them have their answers included in the book, so assigning homework could be a challenge.

The topics are what you expect — the first semester focusing on functions in one variable and the second on functions in several variables. The last chapter is on Fourier series (so the last chapter of the previous edition has been omitted).

New to this edition is an introductory chapter that summarizes what many students taking this class may have learned in a Transition to Higher Mathematics class, such as the principle of mathematical induction, ordered field axioms, and the completeness axioms.

Mikós Bóna is Professor of Mathematics at the University of Florida.

Preface

Part I. ONE-DIMENSIONAL THEORY

1. The Real Number System

1.1 Introduction

1.2 Ordered field axioms

1.3 Completeness Axiom

1.4 Mathematical Induction

1.5 Inverse functions and images

1.6 Countable and uncountable sets

2. Sequences in R

2.1 Limits of sequences

2.2 Limit theorems

2.3 Bolzano-Weierstrass Theorem

2.4 Cauchy sequences

*2.5 Limits supremum and infimum

3. Continuity on R

3.1 Two-sided limits

3.2 One-sided limits and limits at infinity

3.3 Continuity

3.4 Uniform continuity

4. Differentiability on R

4.1 The derivative

4.2 Differentiability theorems

4.3 The Mean Value Theorem

4.4 Taylor's Theorem and l'Hôpital's Rule

4.5 Inverse function theorems

5 Integrability on R

5.1 The Riemann integral

5.2 Riemann sums

5.3 The Fundamental Theorem of Calculus

5.4 Improper Riemann integration

*5.5 Functions of bounded variation

*5.6 Convex functions

6. Infinite Series of Real Numbers

6.1 Introduction

6.2 Series with nonnegative terms

6.3 Absolute convergence

6.4 Alternating series

*6.5 Estimation of series

7. Infinite Series of Functions

7.1 Uniform convergence of sequences

7.2 Uniform convergence of series

7.3 Power series

7.4 Analytic functions

*7.5 Applications

Part II. MULTIDIMENSIONAL THEORY

8. Euclidean Spaces

8.1 Algebraic structure

8.2 Planes and linear transformations

8.3 Topology of Rn

8.4 Interior, closure, boundary

9. Convergence in Rn

9.1 Limits of sequences

9.2 Heine-Borel Theorem

9.3 Limits of functions

9.4 Continuous functions

*9.5 Compact sets

*9.6 Applications

10. Metric Spaces

10.1 Introduction

10.2 Limits of functions

10.3 Interior, closure, boundary

10.4 Compact sets

10.5 Connected sets

10.6 Continuous functions

10.7 Stone-Weierstrass Theorem

11. Differentiability on Rn

11.1 Partial derivatives and partial integrals

11.2 The definition of differentiability

11.3 Derivatives, differentials, and tangent planes

11.4 The Chain Rule

11.5 The Mean Value Theorem and Taylor's Formula

11.6 The Inverse Function Theorem

*11.7 Optimization

12. Integration on Rn

12.1 Jordan regions

12.2 Riemann integration on Jordan regions

12.3 Iterated integrals

12.4 Change of variables

*12.5 Partitions of unity

*12.6 The gamma function and volume

13. Fundamental Theorems of Vector Calculus

13.1 Curves

13.2 Oriented curves

13.3 Surfaces

13.4 Oriented surfaces

13.5 Theorems of Green and Gauss

13.6 Stokes's Theorem

*14. Fourier Series

*14.1 Introduction

*14.2 Summability of Fourier series

*14.3 Growth of Fourier coefficients

*14.4 Convergence of Fourier series

*14.5 Uniqueness

Appendices

A. Algebraic laws

B. Trigonometry

C. Matrices and determinants

E. Vector calculus and physics

F. Equivalence relations

References