# Pedagogy, mathematics and Dieudonné's Foundations of Modern Analysis

I've heard from a friend of mine that Dieudonné's Foundations of Modern Analysis is "painful reading" and "a little outdated"; however, my teacher actually suggested it to me, describing it as a "wonderful text". So I would like to ask here if anyone has gone through it and can describe the general approach followed by the author, and (more importantly) if it is pedagogically and mathematically worthwhile to study analysis using it.

• I read a part of it to understand multiple derivatives of functions of several variables. His treatment of differentiation is fantastic and has insight that you will have difficulty finding in any modern treatment. In short, it is not painful. It is a joy if you appreciate it for what it is. Of course, I never had a course from it, so, what do I know :) – James S. Cook Oct 26 '14 at 3:54
• @JamesS.Cook Thank you for sharing your opinion :) – Dal Oct 26 '14 at 9:23
• You may be interested in this review: projecteuclid.org/download/pdf_1/euclid.bams/1183524143, admittedly published back in 1961. – J W Oct 26 '14 at 14:00
• @JamesS.Cook I had the same experience. I actually developed this understanding myself ($k^{th}$ derivative is a $k$-linear form on tangent space), and was asking around to see if anyone else had developed this perspective. Someone eventually recommended Dieudonne book. It certainly makes the second derivative test a lot more understandable! I actually have a whole online course on this idea here: ximera.osu.edu/course/kisonecat/m2o2c2/course/activity/week1. I thought the idea was pretty enough that it needed wider dissemination. – Steven Gubkin Oct 26 '14 at 14:47
• Yes it is painful, no it is not outdated, no it is not wonderful. I'll elaborate in an answer. – Georges Elencwajg Oct 28 '14 at 20:03

That is the book that had the most deleterious effect on me in my life!

Since the courses in (advanced) calculus I had as an undergraduate were very old-fashioned (emphasis on esoteric criteria for convergence like Raabe's test, nonsensical definition of differentials, ..), a young teaching assistant told me to read Dieudonné's book.
I tried to, but the result was a disaster: I could formally understand the definition of the differential of a map between Banach spaces but I didn't learn that the differential of a map $\mathbb R^2\to \mathbb R^2$ can be represented by a $2$ by $2$ matrix.
One of the first examples in the chapter on metric spaces is $\mathbb Q$ with a $p$-adic distance ...
In the first (!) set of exercises one is requested to prove that in an ultrametric space, given two non-disjoint balls, one is included in the other.
I was completely discouraged, since I couldn't do the exercises, and thought that I would never understand mathematics, but fortunately I stumbled on Buck's Advanced Calculus and fell in love with differntial calculus in several variables.

So my take on Foundations of Modern Analysis is that it is inadequate for learning said foundations and that the ratio abstraction/content is unreasonably high.
Also it contains zero pictures, which I find intolerable.
I would recommend instead Lang's excellent, richly and intelligently illustrated, books on Calculus, Advanced Calculus and Analysis

Warning: the book and the author are not to be confused!
I knew Dieudonné, admired and loved him.
He was one of the great mathematicians of the twentieth century, the main and most energetic contributor to Bourbaki, the collaborator of Grothendieck who would not have written EGA without Dieudonné, the inventor of paracompact spaces and partitions of unity, the inventor of determinants over skew-fields, etc...
Even more importantly, he was selfless, utterly honest and, despite his important position in French academic life, was ready to very humbly put himself at the disposal of young mathematicians like Grothendieck whose talent he helped develop.
I miss the mathematician and the gentleman.

• What did you know at the time? – Billy Rubina Feb 11 '18 at 5:02

Dieudonné's text is very good,but it's definitely not for beginners. In fact, when he wrote the book's original lectures in the late 1950's, the book was intended as a graduate textbook. I doubt it could be used that way today without extensive supplementing-too much progress has been made in analysis.

During the Bourbaki era, there was a lot of chatter among the acolytes of that movement that Foundations would quickly replace Rudin as the standard undergraduate analysis text for serious undergraduates. Despite the golden age of mathematics education and research that was occurring then,I doubt any but the most gifted undergraduates could seriously learn analysis from it.

However, if you like Dieudonné's approach to analysis on normed-particularly Banach-spaces, then I have several excellent recommendations for you.

The definitive treatment of calculus on Banach spaces was authored by Henri Cartan in the mid-1960's in his famous Cours de calcul différentiel at the University of Paris at Sorbonne. Cartan later wrote up the course as 2 volumes that were later translated into English: Differential Calculus and Differential Forms. Together, the 2 books present a concise,rigorous course on differential and integral calculus on Banach spaces. Cartan writes beautifully and it's a terrific,deep and informative course of study.

For a long time, both books were out of print and very difficult to obtain without spending a king's ransom. Forms, the second half on integration on Banach spaces, was republished by Dover in 2006 and is readily and cheaply available-but it's very difficult to read and use without the first half since there aren't a lot of pre-graduate level books on the subject.

I'm happy to report that the first volume has just been republished via Createspace in both an inexpensive paperback and Kindle e-book.

How do I know this? Well, because I'm the publisher. Go,me.........lol

Ok, all kidding aside,I'm serious that my book and its sequel could be just what you're looking for. I'm not only really proud to make this book available again cheaply and readily, I've also included a new detailed preface that describes the historical background of the book in depth and-more importantly for you,I think-a detailed bibliography that suggests not only how to use the books in a course or self-study, but how to use the books as supplements to standard analysis textbooks like Rudin. It also suggests several other books to use in concert with Cartan's books for a complete analysis course on normed spaces. I think you may find several excellent alternatives to Dieudonné's text suggested therein-many of them very inexpensive!

The website of the book can be found with a lot more information here. I hope you'll find it very helpful.These books are classics and should be available to everyone cheaply.

• I love you man. I want this book ever since I bought the Dover and cursed the price of the 1st editions I could find. This is exciting news. I will be looking to buy this. Thanks! – James S. Cook Aug 17 '17 at 1:32
• @JamesS.Cook You want to REALLY thank me,James? Write a short (hopefully positive!) review of the book at the Amazon page.You can also email your colleagues and honor student societies to let them know the 1st volume is finally available again.I'm struggling to get the word out. That will be all the thanks I need. : ) – The Mathemagician Aug 27 '17 at 4:14
• I'll try to do that, I just got it in the mail a couple days ago. The printing of the book is good quality, I am satisfied. – James S. Cook Aug 27 '17 at 13:05
• @JamesS.Cook Really glad to hear it and a review from you at the Amazon page would help get the word out. Even a short positive review would help get the ball rolling. – The Mathemagician Aug 28 '17 at 4:00

According to the author "This volume is an outgrowth of a course intended for first year graduate students or exceptionally advanced undergraduate students ...". The book cannot be recommended for students who don't know the stuff and want to learn it. And it cannot be recommended from general principles: In modern education we endavour to include the student's whole person, i.e., all is senses and capabilities by applying a diversity of media. Dieudonné however does refrain "deliberately from introducing any diagram in the book" to expell geometric intuition. This shows that the author is mentally very unbalanced and one-sided and obviously unable to understand that diagrams as well as movies or spoken text are as well a means of communication and therefore of teaching and thinking as written language. Now, a printed book is excluding spoken text and movies but not diagrams! It is possible to understand him, but it is stupid to rely solely on written text. To think that only the latter is suitable for sharp and axiomatic thinking is somehow as if you willfully blinded yourself in order to adhere solely to some artificial language which is believed to be particularly precise (but is not).

In order to judge the praised Bourbaki approach we should not trust in voices of mathematicians who are as one-sided and restricted in their means of communication as Dieudonné and his collaborators. Here are some voices concerning "Bourbaki" (what a childish idea and name already):

... the Bourbakian abstract nonsense leaves you with such a bitter taste that it feels more like Hell. (Doron Zeilberger)

... mercifully, the Bourbaki plague is dying out. (Murray Gell-Mann)

Only if you are a very one-sided student with no visual abilities this book might be of some help for you.

• I'm very visual, I use lots of diagrammatic thinking. This book was useful to me precisely because its strength is its abstraction. – James S. Cook Aug 18 '17 at 23:14
• The book is not useless. But abstraction should not be confused with abandoning tools of communication which have proved useful for thousands of years. I am always reminded of Blaise Pascal who toward the end of his life wore a spiked belt. – Heinrich Aug 19 '17 at 11:36