**Pure and Applied Undergraduate Texts**

Volume: 18;
2012;
398 pp;
Hardcover

MSC: Primary 26; 03;

Print ISBN: 978-0-8218-8984-8

Product Code: AMSTEXT/18

List Price: $74.00

Individual Member Price: $59.20

**Electronic ISBN: 978-0-8218-9190-2
Product Code: AMSTEXT/18.E**

List Price: $74.00

Individual Member Price: $59.20

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#### Supplemental Materials

# Foundations of Analysis

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*Joseph L. Taylor*

Foundations of Analysis is an excellent new text for undergraduate students in real analysis. More than other texts in the subject, it is clear, concise and to the point, without extra bells and whistles. It also has many good exercises that help illustrate the material. My students were very satisfied with it.

—Nat Smale, University of Utah

I have taught our Foundations of Analysis course (based on Joe Taylor.s book) several times recently, and have enjoyed doing so. The book is well-written, clear, and concise, and supplies the students with very good introductory discussions of the various topics, correct and well-thought-out proofs, and appropriate, helpful examples. The end-of-chapter problems supplement the body of the text very well (and range nicely from simple exercises to really challenging problems).

—Robert Brooks, University of Utah

An excellent text for students whose future will include contact with mathematical analysis, whatever their discipline might be. It is content-comprehensive and pedagogically sound. There are exercises adequate to guarantee thorough grounding in the basic facts, and problems to initiate thought and gain experience in proofs and counterexamples. Moreover, the text takes the reader near enough to the frontier of analysis at the calculus level that the teacher can challenge the students with questions that are at the ragged edge of research for undergraduate students. I like it a lot.

—Don Tucker, University of Utah

My students appreciate the concise style of the book and the many helpful examples.

—W.M. McGovern, University of Washington

Analysis plays a crucial role in the undergraduate curriculum.
Building upon the familiar notions of calculus, analysis introduces
the depth and rigor characteristic of higher mathematics courses.
Foundations of Analysis has two main goals. The first is to
develop in students the mathematical maturity and sophistication they
will need as they move through the upper division curriculum. The
second is to present a rigorous development of both single and several
variable calculus, beginning with a study of the properties of the
real number system.

The presentation is both thorough and concise, with simple,
straightforward explanations. The exercises differ widely in level of
abstraction and level of difficulty. They vary from the simple to the
quite difficult and from the computational to the theoretical. Each
section contains a number of examples designed to illustrate the
material in the section and to teach students how to approach the
exercises for that section.

The list of topics covered is rather standard, although the
treatment of some of them is not. The several variable material makes
full use of the power of linear algebra, particularly in the treatment
of the differential of a function as the best affine approximation to
the function at a given point. The text includes a review of several
linear algebra topics in preparation for this material. In the final
chapter, vector calculus is presented from a modern point of view,
using differential forms to give a unified treatment of the major
theorems relating derivatives and integrals: Green's, Gauss's, and
Stokes's Theorems.

At appropriate points, abstract metric spaces, topological spaces,
inner product spaces, and normed linear spaces are introduced, but
only as asides. That is, the course is grounded in the concrete world
of Euclidean space, but the students are made aware that there are
more exotic worlds in which the concepts they are learning may be
studied.

#### Table of Contents

# Table of Contents

## Foundations of Analysis

- Cover Cover11 free
- Title page i2 free
- Contents iii4 free
- Preface vii8 free
- The real numbers 112 free
- Sequences 3344
- Continuous functions 5970
- The derivative 7990
- The integral 101112
- Infinite series 129140
- Convergence in Euclidean space 161172
- Functions on Euclidean space 191202
- Differentiation in several variables 223234
- Integration in several variables 275286
- Vector calculus 317328
- Degrees of infinity 375386
- Bibliography 387398
- Index 389400 free
- Back Cover Back Cover1410

#### Readership

Undergraduate students interested in real analysis.