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Electronic ISBN:  9781470412135 
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Book Details2005; 151 ppMSC: Primary 26;
This book is written by awardwinning 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 epsilondelta, 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).ReadershipUndergraduate students interested in real analysis.

Table of Contents

Cover

Title

Copyright

Contents

Preface

Part I. Real Numbers and Limits

Chapter 1. Numbers and Logic

Chapter 2. Infinity

Chapter 3. Sequences

Chapter 4. Functions and Limits

Part II. Topology

Chapter 5. Open and Closed Sets

Chapter 6. Continuous Functions

Chapter 7. Composition of Functions

Chapter 8. Subsequences

Chapter 9. Compactness

Chapter 10. Existence of Maximum

Chapter 11. Uniform Continuity

Chapter 12. Connected Sets and the Intermediate Value Theorem

Chapter 13. The Cantor Set and Fractals

Part III. Calculus

Chapter 14. The Derivative and the Mean Value Theorem

Chapter 15. The Riemann Integral

Chapter 16. The Fundamental Theorem of Calculus

Chapter 17. Sequences of Functions

Chapter 18. The Lebesgue Theory

Chapter 19. Infinite Series ∑a[sub(n)]

Chapter 20. Absolute Convergence

Chapter 21. Power Series

Chapter 22. Fourier Series

Chapter 23. Strings and Springs

Chapter 24. Convergence of Fourier Series

Chapter 25. The Exponential Function

Chapter 26. Volumes of nBalls and the Gamma Function

Part IV. Metric Spaces

Chapter 27. Metric Spaces

Chapter 28. Analysis on Metric Spaces

Chapter 29. Compactness in Metric Spaces

Chapter 30. Ascoli's Theorem

Partial Solutions to Exercises

Greek Letters

Index

A

B

C

D

E

F

G

H

I

J

L

M

N

O

P

R

S

T

U

V

W

Back Cover


Additional Material

Reviews

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


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This book is written by awardwinning 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 epsilondelta, 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).
Undergraduate students interested in real analysis.

Cover

Title

Copyright

Contents

Preface

Part I. Real Numbers and Limits

Chapter 1. Numbers and Logic

Chapter 2. Infinity

Chapter 3. Sequences

Chapter 4. Functions and Limits

Part II. Topology

Chapter 5. Open and Closed Sets

Chapter 6. Continuous Functions

Chapter 7. Composition of Functions

Chapter 8. Subsequences

Chapter 9. Compactness

Chapter 10. Existence of Maximum

Chapter 11. Uniform Continuity

Chapter 12. Connected Sets and the Intermediate Value Theorem

Chapter 13. The Cantor Set and Fractals

Part III. Calculus

Chapter 14. The Derivative and the Mean Value Theorem

Chapter 15. The Riemann Integral

Chapter 16. The Fundamental Theorem of Calculus

Chapter 17. Sequences of Functions

Chapter 18. The Lebesgue Theory

Chapter 19. Infinite Series ∑a[sub(n)]

Chapter 20. Absolute Convergence

Chapter 21. Power Series

Chapter 22. Fourier Series

Chapter 23. Strings and Springs

Chapter 24. Convergence of Fourier Series

Chapter 25. The Exponential Function

Chapter 26. Volumes of nBalls and the Gamma Function

Part IV. Metric Spaces

Chapter 27. Metric Spaces

Chapter 28. Analysis on Metric Spaces

Chapter 29. Compactness in Metric Spaces

Chapter 30. Ascoli's Theorem

Partial Solutions to Exercises

Greek Letters

Index

A

B

C

D

E

F

G

H

I

J

L

M

N

O

P

R

S

T

U

V

W

Back Cover

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