Many students acquire knowledge of a large number of theorems and methods of calculus without being able to say how they

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*Table of contents : A Companion to Analysis Title Page Contents Introduction Chapter 1. The Real Line §1.1. Why do we bother? §1.2. Limits §1.3. Continuity §1.4. The fundamental axiom §1.5. The axiom of Archimedes §1.6. Lion hunting §1.7. The mean value inequality §1.8. Full circle §1.9. Are the real numbers unique? Chapter 2. A First Philosophical Interlude §2.1. Is the intermediate value theorem obvious? Chapter 3. Other Versions of the Fundamental Axiom §3.1. The supremum §3.2. The Bolzano—Weierstrass theorem §3.3. Some general remarks Chapter 4. Higher Dimensions §4.1. Bolzano-Weierstrass in Higher Dimensions §4.2. Open and closed sets §4.3. A central theorem of analysis §4.4. The mean value theorem §4.5. Uniform continuity §4.6. The general principle of convergence Chapter 5. Sums and Suchlike §5.1. Comparison tests §5.2. Conditional convergence §5.3. Interchanging limits §5.4. The exponential function §5.5. The trigonometric functions §5.6. The logarithm §5.7. Powers §5.8. The fundamental theorem of algebra Chapter 6. Differentiation §6.1. Preliminaries §6.2. The operator norm and the chain rule §6.3. The mean value inequality in higher dimensions Chapter 7. Local Taylor Theorems §7.1. Some one-dimensional Taylor theorems §7.2. Some many-dimensional local Taylor theorems §7.3. Critical points Chapter 8. The Riemann Integral §8.1. Where is the problem ? §8.2. Riemann integration §8.3. Integrals of continuous functions §8.4. First steps in the calculus of variations §8.5. Vector-valued integrals Chapter 9. Developments and Limitations of the Riemann Integral §9.1. Why go further? §9.2. Improper integrals §9.3. Integrals over areas §9.4. The Riemann- Stieltjes integral §9.5. How long is a piece of string? Chapter 10. Metric Spaces §10.1. Sphere packing §10.2. Shannon's theorem §10.3. Metric spaces §10.4. Norms and the interaction of algebra and analysis §10.5. Geodesics Chapter 11. Complete Metric Spaces §11.1. Completeness §11.2. The Bolzano-Weierstrass property §11.3. The uniform norm §11.4. Uniform convergence §11.5. Power series §11.6. Fourier series Chapter 12. Contraction Mappings and Differential Equations §12.1. Banach's contraction mapping theorem §12.2. Existence of solutions of differential equations §12.3. Local to global §12.4. Green's function solutions Chapter 13. Inverse and Implicit Functions §13.1. The inverse function theorem §13.2. The implicit function theorem §13.3. Lagrange multipliers Chapter 14. Completion §14.1. What is the correct question? §14.2. The solution §14.3. Why do we construct the reals? §14.4. How do we construct the reals? §14.5. Paradise lost? Appendix A. Ordered Fields Appendix B. Countability Appendix C. The Care and Treatment of Counterexamples Appendix D. A More General View of Limits Appendix E. Traditional Partial Derivatives Appendix F. Another Approach to the Inverse Function Theorem Appendix G. Completing Ordered Fields Appendix H. Constructive Analysis Appendix I. Miscellany Appendix J. Executive Summary Appendix K. Exercises Bibliography Index Back CoverPartial Solutions for Questions in Appendix K*

http://dx.doi.org/10.1090/gsm/062

A Companion to Analysis A Second First and First Second Course in Analysis

A Companion to Analysis A Second First and First Second Course in Analysis

T. W. Korner

Graduate Studies in Mathematics Volume 62

a 5

American Mathematical Society Providence, Rhode Island

EDITORIAL

COMMITTEE

Walter Craig Nikolai Ivanov Steven G. Krantz David Saltman (Chair) 2000 Mathematics Subject Classification. Primary 26-01.

For additional information and updates on this book, visit www.ams.org/bookpages/gsm-62

Library of Congress Cataloging-in-Publication D a t a Korner, T. W. (Thomas Williamm), 1946A companion to analysis : a second first and first second course in analysis / T. W. Korner. p. cm. — (Graduate studies in mathematics, ISSN 1065-7339 ; v. 62) Includes bibliographical references and index. ISBN 0-8218-3447-9 (alk. paper) 1. Mathematical analysis. I. Title. II. Series. QA300.K589 515—dc22

2003 2003062905

C o p y i n g and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. © 2004 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

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[Archimedes] concentrated his ambition exclusively upon those speculations which are untainted by the claims of necessity. These studies, he believed, are incomparably superior to any others, since here the grandeur and beauty of the subject matter vie for our admiration with the cogency and precision of the methods of proof. Certainly in the whole science of geometry it is impossible to find more difficult and intricate problems handled in simpler and purer terms than in his works. Some writers attribute it to his natural genius. Others maintain that phenomenal industry lay behind the apparently effortless ease with which he obtained his results. The fact is that no amount of mental effort of his own would enable a man to hit upon the proof of one of Archimedes' theorems, and yet as soon as it is explained to him, he feels he might have discovered it himself, so smooth and rapid is the path by which he leads us to the required conclusion. Plutarch Life of Marcellus [Scott-Kilvert's translation]

It may be observed of mathematicians that they only meddle with such things as are certain, passing by those that are doubtful and unknown. They profess not to know all things, neither do they affect to speak of all things. What they know to be true, and can make good by invincible argument, that they publish and insert among their theorems. Of other things they are silent and pass no judgment at all, choosing rather to acknowledge their ignorance, than affirm anything rashly. Barrow Mathematical Lectures

For [A. N.] Kolmogorov mathematics always remained in part a sport. But when ... I compared him with a mountain climber who made first ascents, contrasting him with I. M. Geffand whose work I compared with the building of highways, both men were offended. ' ... Why, you don't think I am capable of creating general theories?' said Andrei Nikolaevich. 'Why, you think I can't solve difficult problems?' added I. M. V. I. Arnol'd in Kolmogorov in Perspective

A Companion to Analysis

http://dx.doi.org/10.1090/gsm/062

Contents

Introduction Chapter 1. The Real Line §1.1.

Why do we bother?

§1.2.

Limits

§1.3.

Continuity

§1.4.

The fundamental axiom

§1.5.

The axiom of Archimedes

§1.6.

Lion hunting

§1.7.

The mean value inequality

§1.8.

Full circle

§1.9.

Are the real numbers unique?

Chapter 2. §2.1.

A First Philosophical Interlude W

Is the intermediate value theorem obvious?

Chapter 3.

W

Other Versions of the Fundamental Axiom

§3.1.

The supremum

§3.2.

The Bolzano-Weierstrass theorem

§3.3.

Some general remarks

Contents

vm Chapter 4.

Higher Dimensions

43

§4.1.

Bolzano-Weierstrass in Higher Dimensions

43

§4.2.

Open and closed sets

48

§4.3.

A central theorem of analysis

56

§4.4.

The mean value theorem

59

§4.5.

Uniform continuity

64

§4.6.

The general principle of convergence

66

Chapter 5.

Sums and Suchlike V

73

§5.1.

Comparison tests W

73

§5.2.

Conditional convergence