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Mathematical Tapas, Volume 1

Jean-Baptiste Hiriart-Urruty
Publication Date: 
Number of Pages: 
Springer Undergraduate Mathematics Series
Problem Book
[Reviewed by
Allen Stenger
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This book is full of interesting problems. They are at an undergraduate level, and deal primarily with real analysis, with a good bit of linear algebra and matrices and a little bit of number theory mixed in. They are divided into three sections (marked by star ratings), one for each of the first three years of university study. The higher problems are not more difficult, but require more advanced mathematics. A Volume 2 is projected by the same author, that will cover graduate-level mathematics.

The problems are selected so that working each one will teach you something; they are not drill, and do not yield to the obvious standard methods. They are not grouped by subject or in any logical order, but there is a topic index in the front that gives some guidance.

My gold standard for problem books is Pólya & Szegö’s Problems and Theorems in Analysis (PS for short). The present book (JBHU for short), although very good, does not measure up to this standard, in a couple of areas. (1) PS presents the problems in sequences, where the problems in a sequence build on preceding problems. If you work your way through, you will have learned wide swaths of analysis. JBHU’s problems nearly all appear in isolation and there are no links between them. Each one does teach you something, but it’s not as organized as PS. This is probably inevitable, given that most of the problems in JBHU are sourced from journal problem columns such as that of the American Mathematical Monthly. (2) The solutions in JBHU are extremely skimpy and only about half the problems have any solution at all, and there are no references given. PS gives a solution for each problem, although it is often extremely concise, and (except for well-known material) gives a reference to the original source. JBHU does have an intermediate section of hints, and these are very good.

The production of the book leaves something to be desired. The type is small and a little hard to read (it looks like about 8.5 points on 10-point leading). Despite the small type, the top and bottom margins are abnormally large. My guess is that the book was laid out in some different format (perhaps A4 manuscript pages) and then printed smaller rather than being re-flowed for book format. Happily the book seems almost free of typographical errors; I noticed one in Problem 117 that makes the problem incomprehensible, although by peeking ahead to the hints you can figure out the correct reading.

Bottom line: a valuable book that could be better.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is His mathematical interests are number theory and classical analysis.

See the table of contents in the publisher's webpage.