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Hardcover ISBN:  9780821834473 
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Hardcover ISBN:  9780821834473 
Product Code:  GSM/62 
List Price:  $135.00 
MAA Member Price:  $121.50 
AMS Member Price:  $108.00 
eBook ISBN:  9781470421052 
Product Code:  GSM/62.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Hardcover ISBN:  9780821834473 
eBook ISBN:  9781470421052 
Product Code:  GSM/62.B 
List Price:  $220.00 $177.50 
MAA Member Price:  $198.00 $159.75 
AMS Member Price:  $176.00 $142.00 

Book DetailsGraduate Studies in MathematicsVolume: 62; 2004; 590 ppMSC: Primary 26;
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 finitedimensional 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, wellknown author Thomas Körner provides able and hardworking students a great text for independent study or for an advanced undergraduate or firstlevel graduate course. It includes many stimulating exercises. An appendix contains a large number of accessible but nonroutine problems that will help students advance their knowledge and improve their technique.
ReadershipAdvanced undergraduates, graduate students and research mathematicians interested in analysis.

Table of Contents

Chapters

Chapter 1. The real line

Chapter 2. A first philosophical interlude

Chapter 3. Other versions of the fundamental axiom

Chapter 4. Higher dimensions

Chapter 5. Sums and suchlike

Chapter 6. Differentiation

Chapter 7. Local Taylor theorems

Chapter 8. The Riemann integral

Chapter 9. Developments and limitations of the Riemann integral

Chapter 10. Metric spaces

Chapter 11. Complete metric spaces

Chapter 12. Contraction mappings and differential equations

Chapter 13. Inverse and implicit functions

Chapter 14. Completion

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


Additional Material

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 A Companion to Analysis is a welcome addition to the literature on undergraduatelevel rigorous analysis. It is written with great care with regard to both mathematical correctness and pedagogical soundness. Körner shows good taste in deciding what to explain in detail and what to leave to the reader in the exercises scattered throughout the text. And the enormous collection of supplementary exercises in Appendix K, which comprises almost onethird of the whole book, is a valuable resource for both teachers and students.
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


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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 finitedimensional 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, wellknown author Thomas Körner provides able and hardworking students a great text for independent study or for an advanced undergraduate or firstlevel graduate course. It includes many stimulating exercises. An appendix contains a large number of accessible but nonroutine problems that will help students advance their knowledge and improve their technique.
Advanced undergraduates, graduate students and research mathematicians interested in analysis.

Chapters

Chapter 1. The real line

Chapter 2. A first philosophical interlude

Chapter 3. Other versions of the fundamental axiom

Chapter 4. Higher dimensions

Chapter 5. Sums and suchlike

Chapter 6. Differentiation

Chapter 7. Local Taylor theorems

Chapter 8. The Riemann integral

Chapter 9. Developments and limitations of the Riemann integral

Chapter 10. Metric spaces

Chapter 11. Complete metric spaces

Chapter 12. Contraction mappings and differential equations

Chapter 13. Inverse and implicit functions

Chapter 14. Completion

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

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 A Companion to Analysis is a welcome addition to the literature on undergraduatelevel rigorous analysis. It is written with great care with regard to both mathematical correctness and pedagogical soundness. Körner shows good taste in deciding what to explain in detail and what to leave to the reader in the exercises scattered throughout the text. And the enormous collection of supplementary exercises in Appendix K, which comprises almost onethird of the whole book, is a valuable resource for both teachers and students.
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