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A Companion to Analysis: A Second First and First Second Course in Analysis
T. W. Körner, University of Cambridge, England

Graduate Studies in Mathematics
2004; 590 pp; hardcover
Volume: 62
ISBN-10: 0-8218-3447-9
ISBN-13: 978-0-8218-3447-3
List Price: US$89
Member Price: US$71.20
Order Code: GSM/62
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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 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.


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


"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 undergraduate-level 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 one-third 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

Table of Contents

  • The real line
  • A first philosophical interlude \(\heartsuit\heartsuit\)
  • Other versions of the fundamental axiom
  • Higher dimensions
  • Sums and suchlike \(\heartsuit\)
  • Differentiation
  • Local Taylor theorems
  • The Riemann integral
  • Developments and limitations of the Riemann integral \(\heartsuit\)
  • Metric spaces
  • Complete metric spaces
  • Contraction mappings and differential equations
  • Inverse and implicit functions
  • Completion
  • Ordered fields
  • Countability
  • The care and treatment of counterexamples
  • A more general view of limits
  • Traditional partial derivatives
  • Another approach to the inverse function theorem
  • Completing ordered fields
  • Constructive analysis
  • Miscellany
  • Executive summary
  • Exercises
  • Bibliography
  • Index
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