The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities 1st Edition by J. Michael Steele (PDF)

Description

This lively, problem-oriented text, first published in 2004, is designed to coach readers toward mastery of the most fundamental mathematical inequalities. With the Cauchy-Schwarz inequality as the initial guide, the reader is led through a sequence of fascinating problems whose solutions are presented as they might have been discovered – either by one of history’s famous mathematicians or by the reader. The problems emphasize beauty and surprise, but along the way readers will find systematic coverage of the geometry of squares, convexity, the ladder of power means, majorization, Schur convexity, exponential sums, and the inequalities of Hölder, Hilbert, and Hardy. The text is accessible to anyone who knows calculus and who cares about solving problems. It is well suited to self-study, directed study, or as a supplement to courses in analysis, probability, and combinatorics.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐In many respects this is one of my very favorite math books. I’m tempted to say it is simply the best in a lot of ways. The worked problems and explanations in the chapters are very high quality, and the exercises (with thoughtful solutions) are very good. As the author notes, there is a lot of variety in the book, making it a quality companion for a course in analysis, probability or combinatorics. Plus inequalities don’t seem to be taught very well, and this book really does ‘teach’ you them.I dinged this one star because the book has the thickest errata sheet I’ve ever seen. There are even comments in the errata sheet (for example regarding ex 6.8) saying that a problem is fatally flawed and the author plans to update it in 2007. A decade passes and still no update or re-prints of the book.Still, all things considered, I love this book.- – – -edit: I decided to upgrade this to 5 stars. Even a few months after finishing it, I find myself re-visiting a few topics in the book and liking it even more. It really is outstanding.

⭐Prof. Steele has done a great job in providing an “entertaining” (if I may say) book on inequalities. Along with Cauchy-Schwarz inequality the book provides very “lively and problem oriented” (adjectives from the first page of the book) chapters that are insightful and enjoyable. For example, the way you are introduced to Cauchy-Schwarz inequality involves attempting it as a “problem” – first looking at cases n=1 and n=2 trying to use induction. After that you get into fancy approach using quadratic expression. Such an approach throughout the book makes this book really enjoyable. Solutions provided make it ideal for self learning as well as a book to entertain yourself when you get bored 🙂

⭐A wonderful expository book on building brain muscle for mathematical proof–obviously on inequalities. Wish all textbooks are written with such wit and care for the readers!

⭐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!

⭐Professor Steele has done a wonderful job in developing the theory behind the Cauchy-Schwarz inequality. He starts off with the basic theory and then through the course of the book he teases out the limitless ways the inequality can be used. There is a breathtaking sweep of applications. What is interesting and valuable about his approach is that as he develops the building blocks he explains why or why not a particular approach might not work. I think there is quite a bit of Polya’s inspiration in his approach. For instance, he gives Polya’s proof of the Carleman inequality which, on it face, is almost outrageously unbelievable ( where does the “e” come from?) but by that stage you worked through the challenge problems and the other material and it is possible to see why the “e” makes sense.The challenge problems are excellent and his solutions sometimes skip over some important steps which a teacher could get students to fill in so that they can demonstrate that they understand the material.There is a lot to learn from this book and it should be read by everyone who is seriously interested in mathematics. The classic Hardy-Littlewood-Polya book on inequalities is a quite different beast but the two together provide the serious reader with a depth of understanding that is hard to surpass.

⭐This book’s topic, mathematical inequalities, could be considered too esoteric to justify a book for more than a small niche audience of mathematicians. Yet this is an exceptionally well-written book that should appeal not only to those who might need inequalities in their work, but also to any student of mathematics who wants to learn how to discover and present elegant proofs. The only prerequisites are a familiarity with series, sequences, and standard notation in calculus and linear algebra.In addition to its core content, the book does something that too few books in mathematics do: Provide a solution for every exercise. This makes it a precious resource for independent study.

⭐This is a beautifully written book, taking the reader through the approaches and techniques with thoughtful exercises. Time and again I stopped to relish the insights…I forgive the typos! In fact they just keep you on your toes!

⭐A great book. One of the best I have ever read. Let me offer you a physicist’s perspective on it. In physics, we often handle hard problems with ad-hoc approximations based on certain intuitions. This approach is in many cases very instructive, but, inevitably, we loose rigor, and not rarely the approximations go out of control. Providing bounds could often be as illuminating as an approximation based on physical arguments, while at the same time rigor is retained. How to give such bounds has to be learned, however. This book offers an excellent introduction on how to think in terms of bounds and how to develop strategies for finding them. Highly recommendable.

⭐Those inequalities screwed me over and over again back in university, but now taking another look at those, feel like seeing some good old friends :)This book is very well-written. I’ve also tried solving some exercises while reading it, fairly challenging but also entertaining.Overall, highly recommended!

⭐e’ una vera miniera di tesori sull’argomento.lo consiglio a chiunque e’ appassionato alla matematica e in particolare alla bellezza argomentativa delle prove matematiche.

⭐

⭐it has a nice, motivating atmosphere, and gives you a kind of “you are special!” feeling :)The exercises are very well constructed and perfectly placed.

Keywords

Free Download The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities 1st Edition in PDF format
The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities 1st Edition PDF Free Download
Download The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities 1st Edition 2004 PDF Free
The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities 1st Edition 2004 PDF Free Download
Download The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities 1st Edition PDF
Free Download Ebook The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities 1st Edition