**Graduate Studies in Mathematics**

Volume: 62;
2004;
590 pp;
Hardcover

MSC: Primary 26;

Print ISBN: 978-0-8218-3447-3

Product Code: GSM/62

List Price: $89.00

Individual Member Price: $71.20

**Electronic ISBN: 978-1-4704-2105-2
Product Code: GSM/62.E**

List Price: $89.00

Individual Member Price: $71.20

#### Supplemental Materials

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

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*T. W. Körner*

Many students acquire knowledge of a large number of theorems and
methods of calculus without being able to say how they work together. This book
provides those students with the coherent account that they need. A
Companion to Analysis explains the problems that must be resolved in order
to procure a rigorous development of the calculus and shows the student how to
deal with those problems.

Starting with the real line, the book moves on to finite-dimensional
spaces and then to metric spaces. Readers who work through this text will be
ready for courses such as measure theory, functional analysis, complex
analysis, and differential geometry. Moreover, they will be well on the road
that leads from mathematics student to mathematician.

With this book, well-known author Thomas Körner provides able and
hard-working students a great text for independent study or for an advanced
undergraduate or first-level graduate course. It includes many stimulating
exercises. An appendix contains a large number of accessible but non-routine
problems that will help students advance their knowledge and improve their
technique.

#### Table of Contents

# Table of Contents

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

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents vii8 free
- Introduction xi12 free
- Chapter 1. The Real Line 116 free
- Chapter 2. A First Philosophical Interlude 2540
- Chapter 3. Other Versions of the Fundamental Axiom 3146
- Chapter 4. Higher Dimensions 4358
- Chapter 5. Sums and Suchlike 7388
- Chapter 6. Differentiation 117132
- Chapter 7. Local Taylor Theorems 135150
- Chapter 8. The Riemann Integral 161176
- Chapter 9. Developments and Limitations of the Riemann Integral 195210
- Chapter 10. Metric Spaces 221236
- Chapter 11. Complete Metric Spaces 249264
- Chapter 12. Contraction Mappings and Differential Equations 287302
- Chapter 13. Inverse and Implicit Functions 311326
- Chapter 14. Completion 335350
- Appendix A. Ordered Fields 357372
- Appendix B. Countability 361376
- Appendix C. The Care and Treatment of Counterexamples 365380
- Appendix D. A More General View of Limits 371386
- Appendix E. Traditional Partial Derivatives 377392
- Appendix F. Another Approach to the Inverse Function Theorem 383398
- Appendix G. Completing Ordered Fields 387402
- Appendix H. Constructive Analysis 391406
- Appendix I. Miscellany 395410
- Appendix J. Executive Summary 401416
- Appendix K. Exercises 405420
- Bibliography 583598
- Index 585600
- Back Cover Back Cover1608

#### Readership

Advanced undergraduates, graduate students and research mathematicians interested in analysis.

#### Reviews

This book not only provides a lot of solid information about real analysis, it also answers those questions which students want to ask but cannot figure how to formulate. To read this book is to spend time with one of the modern masters in the subject.

-- Steven G. Krantz, Washington University, St. Louis

T. W. Körner's

One of the major assets of the book is Körner's very personal writing style. By keeping his own engagement with the material continually in view, he invites the reader to a similarly high level of involvement. And the witty and erudite asides that are sprinkled throughout the book are a real pleasure.

-- Gerald Folland, University of Washington, Seattle

This is a remarkable book. It provides deep and invaluble insight into many parts of analysis, presented by an accomplished analysist. Korner covers all of the important aspects of an advanced calculus course along with a discussion of other interesting topics.

-- Paul Sally, University of Chicago

The book is a very useful companion to standard analysis textbooks. It stands out in virtue of the author's style of writing, characterized by a pleasant mixture of various erudite reflections.

-- MAA Reviews