2005;
151 pp;
Hardcover

MSC: Primary 26;

**Print ISBN: 978-0-8218-3670-5
Product Code: REAL**

List Price: $49.00

AMS Member Price: $39.20

MAA Member Price: $44.10

**Electronic ISBN: 978-1-4704-1213-5
Product Code: REAL.E**

List Price: $46.00

AMS Member Price: $36.80

MAA Member Price: $41.40

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#### Supplemental Materials

# Real Analysis

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*Frank Morgan*

This book is written by award-winning author, Frank Morgan. It offers
a simple and sophisticated point of view, reflecting Morgan's insightful
teaching, lecturing, and writing style.

Intended for undergraduates studying real analysis, this book builds the
theory behind calculus directly from the basic concepts of real numbers,
limits, and open and closed sets in \(\mathbb{R}^n\). It gives the three
characterizations of continuity: via epsilon-delta, sequences, and open sets.
It gives the three characterizations of compactness: as "closed and bounded,"
via sequences, and via open covers. Topics include Fourier series, the Gamma
function, metric spaces, and Ascoli's Theorem.

This concise text not only provides efficient proofs, but also shows
students how to derive them. The excellent exercises are accompanied by select
solutions. Ideally suited as an undergraduate textbook, this complete book on
real analysis will fit comfortably into one semester.

Frank Morgan received the first Haimo Award for distinguished college
teaching from the
Mathematical Association of America. He has also garnered top teaching awards
from Rice University (Houston, TX) and MIT (Cambridge, MA).

#### Readership

Undergraduate students interested in real analysis.

#### Reviews & Endorsements

Reading your book is a refreshingly delightful change from the usual emphasis on series, rather than topology, as a foundation of analysis.

-- Robert Jones, University of Dusseldorf

#### Table of Contents

# Table of Contents

## Real Analysis

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Preface vii8 free
- Part I. Real Numbers and Limits 110 free
- Part II. Topology 2736
- Chapter 5. Open and Closed Sets 2938
- Chapter 6. Continuous Functions 3544
- Chapter 7. Composition of Functions 3746
- Chapter 8. Subsequences 3948
- Chapter 9. Compactness 4352
- Chapter 10. Existence of Maximum 4756
- Chapter 11. Uniform Continuity 4958
- Chapter 12. Connected Sets and the Intermediate Value Theorem 5362
- Chapter 13. The Cantor Set and Fractals 5766

- Part III. Calculus 6372
- Chapter 14. The Derivative and the Mean Value Theorem 6574
- Chapter 15. The Riemann Integral 6978
- Chapter 16. The Fundamental Theorem of Calculus 7584
- Chapter 17. Sequences of Functions 7988
- Chapter 18. The Lebesgue Theory 8594
- Chapter 19. Infinite Series ∑a[sub(n)] 8998
- Chapter 20. Absolute Convergence 93102
- Chapter 21. Power Series 97106
- Chapter 22. Fourier Series 103112
- Chapter 23. Strings and Springs 109118
- Chapter 24. Convergence of Fourier Series 113122
- Chapter 25. The Exponential Function 115124
- Chapter 26. Volumes of n-Balls and the Gamma Function 119128

- Part IV. Metric Spaces 123132
- Partial Solutions to Exercises 141150
- Greek Letters 151160
- Index 153162
- Back Cover Back Cover1165