What do you think?
Rate this book
The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities
Michael Steele describes the fundamental topics in mathematical inequalities and their uses. Using the Cauchy-Schwarz inequality as a guide, Steele presents a fascinating collection of problems related to inequalities and coaches readers through solutions, in a style reminiscent of George Polya, by teaching basic concepts and sharpening problem solving skills at the same time. Undergraduate and beginning graduate students in mathematics, theoretical computer science, statistics, engineering, and economics will find the book appropriate for self-study.
318 pages, Paperback
First published January 1, 2004
Ratings & Reviews
Friends & Following
Create a free account to discover what your friends think of this book!
Community Reviews
Displaying 1 - 5 of 5 reviews
September 25, 2013
The Cauchy-Schwarz Master Class is perhaps amongst the best mathematics book that I have seen in many years. True to its name, it is indeed a Master Class. I came across this book 4-5 months ago purely by accident in a bookstore. Sat there and casually read the first chapter and within 30 minutes I was hooked! I regret not coming across it earlier. The author has a rare talent for exposition, replete with interesting historical digressions and very inviting challenge problems. You can literally feel the author's enthusiasm for inequalities while reading the book, and most importanly; he manages to infect you with it! I could not put aside the book completely once I had picked it up and eventually decided to go all the way, slowly over the last 4-5 months (still left with two chapters and some problems). The problems are addictive. Often, when I did pick up the book, I found myself doing nothing but thinking about the exercises I was trying to solve.
Like most great books, the way it is organized makes it "very natural" to rediscover many susbtantial results (some of them named) appearing much later by yourself, provided you happen to just ask the right questions. I believe that this is the sole trademark of a truly remarkable book. This happened with me quite a few times. However, I would like to recount a particular example. I was tutoring a freshman in Linear Algebra around the time I bought the book. I mentioned the book to him and it eventually so happened that I lent it to him for a week. He was stuck with exercise 1.6 (which is an innocuous inequality at first glance). He eventually managed to solve it without hints. However, not only did he manage to solve it but using an insight from there he was able to ask the right question - what happens when you replace the second power with something else? What can you replace it with? In essence he was able to take 1.6 (in which the powers summed to one and this was mentioned) and prove a version of Holder's just by using the inductive proof described in the prior chapter (he hadn't heard of Holder's). I was equally amazed when I was able to formulate and prove some inequalities that actually appeared later in the book.
The book emphasizes a problem solving approach and features a large number of inequalities (while also relating them all the time) which makes sure you make very good friends with some of the most interesting inequalities. Like mentioned earlier, the exercise questions are very well chosen: For example, in the first chapter an exercise (not too hard once one has worked through the challenge problems) is proving the Cramer-Rao lower bound, a cornerstone of modern statistics. Another remarkable example is a "defect form" of Cauchy-Schwarz that is a central component in the proof of the Szemeredi Regularity Lemma, one of the most fundamental results in Graph Theory. All these examples are remarkably provable after reading and working out challenge problems. Steele also often stuns in his digressions. For example: There is a part when the goal is to derive Lagrange's Identity. We move to establish this by trying to "measure" the defect in Cauchy-Schwarz (with is a polynomial). We soon show that this polynomial can be expressed as a sum of squares (and is thus always non-negative). Then we look at Minkowski's conjecture that tries to ask if non-negativity of a polynomial always implies a sum-of-squares. We then learn that it is not possible to do this. However a simple modification to this is Hilbert's 17th problem!
The book starts off with the inequalities dealing with "natural" notions such as monotonicity and positivity (which appear very frequently in Olympiads) and later builds onto somewhat less natural and more advanced notions such as convexity. The book also manages to convey a sense of appreciation of why Cauchy-Schwarz is such a fundamental inequality (by relating it to many different notions such as isometry, isoperimetric inequalities, convexity etc etc). It is a little strange that Cauchy-Schwarz keeps appearing all the time. What makes it so useful and fundamental is indeed quite interesting and non-obvious. It is also not at all clear why is it that it is Cauchy-Schwarz which is mainly useful.
I can't recommend this book enough. It is truly a gem!
Like most great books, the way it is organized makes it "very natural" to rediscover many susbtantial results (some of them named) appearing much later by yourself, provided you happen to just ask the right questions. I believe that this is the sole trademark of a truly remarkable book. This happened with me quite a few times. However, I would like to recount a particular example. I was tutoring a freshman in Linear Algebra around the time I bought the book. I mentioned the book to him and it eventually so happened that I lent it to him for a week. He was stuck with exercise 1.6 (which is an innocuous inequality at first glance). He eventually managed to solve it without hints. However, not only did he manage to solve it but using an insight from there he was able to ask the right question - what happens when you replace the second power with something else? What can you replace it with? In essence he was able to take 1.6 (in which the powers summed to one and this was mentioned) and prove a version of Holder's just by using the inductive proof described in the prior chapter (he hadn't heard of Holder's). I was equally amazed when I was able to formulate and prove some inequalities that actually appeared later in the book.
The book emphasizes a problem solving approach and features a large number of inequalities (while also relating them all the time) which makes sure you make very good friends with some of the most interesting inequalities. Like mentioned earlier, the exercise questions are very well chosen: For example, in the first chapter an exercise (not too hard once one has worked through the challenge problems) is proving the Cramer-Rao lower bound, a cornerstone of modern statistics. Another remarkable example is a "defect form" of Cauchy-Schwarz that is a central component in the proof of the Szemeredi Regularity Lemma, one of the most fundamental results in Graph Theory. All these examples are remarkably provable after reading and working out challenge problems. Steele also often stuns in his digressions. For example: There is a part when the goal is to derive Lagrange's Identity. We move to establish this by trying to "measure" the defect in Cauchy-Schwarz (with is a polynomial). We soon show that this polynomial can be expressed as a sum of squares (and is thus always non-negative). Then we look at Minkowski's conjecture that tries to ask if non-negativity of a polynomial always implies a sum-of-squares. We then learn that it is not possible to do this. However a simple modification to this is Hilbert's 17th problem!
The book starts off with the inequalities dealing with "natural" notions such as monotonicity and positivity (which appear very frequently in Olympiads) and later builds onto somewhat less natural and more advanced notions such as convexity. The book also manages to convey a sense of appreciation of why Cauchy-Schwarz is such a fundamental inequality (by relating it to many different notions such as isometry, isoperimetric inequalities, convexity etc etc). It is a little strange that Cauchy-Schwarz keeps appearing all the time. What makes it so useful and fundamental is indeed quite interesting and non-obvious. It is also not at all clear why is it that it is Cauchy-Schwarz which is mainly useful.
I can't recommend this book enough. It is truly a gem!
April 3, 2017
It's very difficult to teach advanced topics in mathematics in a problem + solution format. Professor Steele has done an extraordinary job of teaching mathematical inequalities in this enjoyable, interactive form.
March 12, 2024
The Cauchy-Schwarz Master Class is a hidden Gem.
Steele's passion for abstraction and simplicity is extremely rare among mathematicians.
Highly recommend for those who are interested in mathematics.
Steele's passion for abstraction and simplicity is extremely rare among mathematicians.
Highly recommend for those who are interested in mathematics.
May 25, 2016
This book teaches you how to think about proving various inequalities.
April 19, 2017
This should be required undergraduate reading material.
Displaying 1 - 5 of 5 reviews