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. 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 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 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
