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A Companion to Analysis: A Second First and First Second Course in Analysis
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.
- GenresMathematicsTextbooks
590 pages, Hardcover
First published January 1, 2004
3 people are currently reading
78 people want to read
About the author
T.W. Körner
13 books9 followersThomas William Körner (born 17 February 1946) is a British pure mathematician and the author of school books. He is titular Professor of Fourier Analysis in the University of Cambridge and a Fellow of Trinity Hall. He is the son of the philosopher Stephan Körner and of Edith Körner.
He studied at Trinity Hall, Cambridge, and wrote his PhD thesis Some Results on Kronecker, Dirichlet and Helson Sets there in 1971, studying under Nicholas Varopoulos. In 1972 he won the Salem Prize.
He has written three academic mathematics books aimed at undergraduates, and two books aimed at secondary school students, the popular 1996 title The Pleasures of Counting and Naive Decision Making (published 2008) on probability, statistics and game theory.
He studied at Trinity Hall, Cambridge, and wrote his PhD thesis Some Results on Kronecker, Dirichlet and Helson Sets there in 1971, studying under Nicholas Varopoulos. In 1972 he won the Salem Prize.
He has written three academic mathematics books aimed at undergraduates, and two books aimed at secondary school students, the popular 1996 title The Pleasures of Counting and Naive Decision Making (published 2008) on probability, statistics and game theory.
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Displaying 1 - 2 of 2 reviews
July 1, 2009
Truly great self-study guide to analysis. This guy has a great voice, but still writes clear math prose.
November 30, 2025
"A Companion to Analysis" by T.W. Körner (pronounced "kerner" I assume) is a "supplemental" analysis book for undergraduates who've done a typical first course. The point of the book seems to be to emphasise ideas which Körner feels undergraduates don't appreciate enough.
For example, the book begins with a demonstration that continuity is meaningless in ℚ, the field of rational numbers, by showing that it is possible to define a function from ℚ to ℚ with a "jump" at √2, and yet still have a "continuous" function by the usual definition of continuity. The author has strong opinions about almost everything, such as the mean value inequality, and he wants to tell you them.
The book continues first with a "philosophical interlude" which the author marks with heart symbols ♡♡, a practice which I think should be discouraged in general. It then goes on to discuss other versions of the fundamental axiom, then into multivariable calculus in chapter 4, and then "Sums and suchlike" which the author again marks with his beloved heart symbol.
The next three chapters are on differentiation, Taylor theorems, and Riemann integration, where Körner emphasises the mathematical difficulties of defining area using examples of Banach, Tarski, and Vitali. Next is a chapter on different kinds of issues with Riemann integration, and then chapter ten goes into metric spaces, with hearted sections on sphere packing and Shannon's theorem from information theory. Chapter 11 goes into complete metric spaces, twelve goes into contraction mappings and differential equations, then chapter 13 gives the inverse and implicit function theorems. Chapter 14 "Completion" consists of a construction of the reals using Cauchy sequences, unlike Spivak's Calculus or Rudin's Principles of Mathematical Analysis which both use Dedekind cuts.
After this come numerous appendices, which since this is a "Companion to Analysis" might actually just as well have been chapters, discussing various issues and giving opinions.
The text contains many "proofs left to the reader", without the dread words themselves, disguised as very easy exercises, often broken down into chunks using roman numerals (i), (ii), (iii) and so on. Answers to these exercises are not supplied. There are no problem sets at the end of chapters but about a quarter of the length of the book is dedicated to the huge appendix K from page 405 to 582, which consists of about 350 exercises of difficulty varying from straightforward to difficult. Körner offers answers to these online at his web page at Cambridge University. There is also a page of typos and other minor errors at Körner's web pages, although for some reason I have not been able to get Körner to respond to some of the things I found in the book. As an "exercise for the reader" you might want to print out his sheet of typos and see if you can find them yourself. I haven't read everything in the book, but so far I have only missed one.
Körner was a professor at the Trinity Hall college of Cambridge University, now emeritus, and apparently his father was a philosopher who studied mathematics and wrote a book on the philosophy of mathematics, which may explain his taste for "philosophical interludes". He has also written books such as The Pleasures of Counting which is intended for prospective mathematics students to read, as well as some monographs. Incidentally the bibliography of this book is a bit thin on examples of books similar to Korner's, so it may be worth looking at The Pleasures of Counting which contains a few suggestions.
I read this book initially because I got stuck on many of the problems in the later chapters of Rudin's "Principles of Mathematical Analysis" and was looking for something a bit easier. I'm very glad to have found this entertaining book and would recommend it.
For example, the book begins with a demonstration that continuity is meaningless in ℚ, the field of rational numbers, by showing that it is possible to define a function from ℚ to ℚ with a "jump" at √2, and yet still have a "continuous" function by the usual definition of continuity. The author has strong opinions about almost everything, such as the mean value inequality, and he wants to tell you them.
The book continues first with a "philosophical interlude" which the author marks with heart symbols ♡♡, a practice which I think should be discouraged in general. It then goes on to discuss other versions of the fundamental axiom, then into multivariable calculus in chapter 4, and then "Sums and suchlike" which the author again marks with his beloved heart symbol.
The next three chapters are on differentiation, Taylor theorems, and Riemann integration, where Körner emphasises the mathematical difficulties of defining area using examples of Banach, Tarski, and Vitali. Next is a chapter on different kinds of issues with Riemann integration, and then chapter ten goes into metric spaces, with hearted sections on sphere packing and Shannon's theorem from information theory. Chapter 11 goes into complete metric spaces, twelve goes into contraction mappings and differential equations, then chapter 13 gives the inverse and implicit function theorems. Chapter 14 "Completion" consists of a construction of the reals using Cauchy sequences, unlike Spivak's Calculus or Rudin's Principles of Mathematical Analysis which both use Dedekind cuts.
After this come numerous appendices, which since this is a "Companion to Analysis" might actually just as well have been chapters, discussing various issues and giving opinions.
The text contains many "proofs left to the reader", without the dread words themselves, disguised as very easy exercises, often broken down into chunks using roman numerals (i), (ii), (iii) and so on. Answers to these exercises are not supplied. There are no problem sets at the end of chapters but about a quarter of the length of the book is dedicated to the huge appendix K from page 405 to 582, which consists of about 350 exercises of difficulty varying from straightforward to difficult. Körner offers answers to these online at his web page at Cambridge University. There is also a page of typos and other minor errors at Körner's web pages, although for some reason I have not been able to get Körner to respond to some of the things I found in the book. As an "exercise for the reader" you might want to print out his sheet of typos and see if you can find them yourself. I haven't read everything in the book, but so far I have only missed one.
Körner was a professor at the Trinity Hall college of Cambridge University, now emeritus, and apparently his father was a philosopher who studied mathematics and wrote a book on the philosophy of mathematics, which may explain his taste for "philosophical interludes". He has also written books such as The Pleasures of Counting which is intended for prospective mathematics students to read, as well as some monographs. Incidentally the bibliography of this book is a bit thin on examples of books similar to Korner's, so it may be worth looking at The Pleasures of Counting which contains a few suggestions.
I read this book initially because I got stuck on many of the problems in the later chapters of Rudin's "Principles of Mathematical Analysis" and was looking for something a bit easier. I'm very glad to have found this entertaining book and would recommend it.
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