A review of my book A First Course in Functional Analysis

A review for my book A First Course in Functional Analysis appeared in Zentralblatt Math – here is a link to the review. I am quite thankful that someone has read my book and bothered to write a review, and that zBMath publishes reviews. That’s all great. Now I have a few words to say about it. This is an opportunity for me to bring up the subject of my book and highlight some things worth highlighting.

I am not too happy about this review. It is not that it is a negative review – actually it has a rather kind air to it. However, I am somewhat disappointed in the information that the review contains, and I am not sure that it does the reader some service which the potential readers could not achieve by simply reading the table of contents and the preface to the book (it is easy to look inside the book in the Amazon page; of course, it is also easy to find a copy of the book online).

The reviewer correctly notices that one key feature of the book is the treatment of as a completion of , and that this is used for applications in analysis. However, I would love it if a reviewer would point out to the fact that, although the idea of thinking about as a completion space is not new, few (if any) have attempted to actually walk the extra mile and work with in this way (i.e., without requiring measure theory) all the way up to rigorous and significant applications in analysis. Moreover, it would be nice if my attempt was compared to other such attempts (if they exist), and I would like to hear opinions about whether my take is successful.

I am grateful that the reviewer reports on the extensive exercises (this is indeed, in my opinion, one of the pluses of new books in general and my book in particular), but there are a couple of other innovations that are certainly worth remarking on, and I hope that the next reviewer does not miss them. For example, is it a good idea to include a chapter on Hilbert function spaces in an introductory text to FA? (a colleague of mine told me that he would keep that out). Another example: I think that my chapter on applications of compact operators is quite special. This chapter has two halves: one on integral equations and one on functional equations. Now, the subject of integral equations is well trodden and takes a central place in some introductions to FA, and one might wonder whether anything new can be done here in terms of the organization and presentation of the material. So, I think it is worth remarking about whether or not my exposition has anything to add. The half on applications of compact operators to functional equations contains some beautiful and highly non-trivial material that has never appeared in a book before, not to mention that functional equations of any kind are rarely considered in introductions to FA; this may also be worth a comment.

A First Course in Functional Analysis (my book)

She’hechiyanu Ve’kiyemanu!

My book, A First Course in Functional Analysis, to be published with Chapman and Hall/CRC, will soon be out. There is already a cover, check it out on the CRC Press website.

This book is written to accompany an undergraduate course in functional analysis, where the course I had in mind is precisely the course that we give here at the Technion, with the same constraints. Constraint number 1: a course in measure theory is not mandatory in our undergraduate program. So how can one seriously teach functional analysis with significant applications? Well, one can, and I hope that this book proves that one can. I already wrote before, measure theory is not a must. Of course anyone going for a graduate degree in math should study measure theory (and get an A), but I’d like the students to be able to study functional analysis before that (so that they can do a masters degree in operator theory with me).

I believe that the readers will find many other original organizational contributions to the presentation of functional analysis in this book, but I leave them for you to discover. Instructors can request an e-copy for inspection (in the link to the publisher website above), friends and direct students can get a copy from me, and I hope that the rest of the world will recommend this book to their library (or wait for the libgen version).

Hal-moss, not Hal-mush

The title of this post is a small service to Paul Halmos. I recently read his book “I Want to be a Mathematician”, subtitled “an Automathography”, where I found this:

Do all readers know that I reject ‘Hal-mush’ – some people’s notion of the “right” way to pronounce me? Please, please, say ‘Hal-moss’.

Sure, as you wish (I occasionally used to say “Hal-mosh”. No more).

Halmos was an influential mathematician who was born a hundred years ago, and died ten years ago. He worked in several areas (measure and ergodic theory, logic, operator theory) and wrote many successful books. He is considered to be a superb expositor. [His Hilbert Space Problem Book is the most refreshing, provocative and captivating book that I ever found accidentally on the library shelf (browsing with no definite goal, when I was a TA in a course on functional analysis). A Hilbert Space Problem Book is not only a beautiful and original idea, it is also executed to perfection and thus very useful.]

Perhaps I will take the opportunity of his 100 hundredth birthday (a few months from now) to write about one or some of his classic papers. But now I want to write about the book “I Want to be a Mathematician”.

The book (the automathography) is a kind of professional autobiography, omitting almost everything in personal life, and concentrating on his life as a mathematician, and that includes almost every aspect of the profession. Halmos has some interesting and definite opinions on various matters, and he believes that they should be expressed unequivocally: “I must not waffle and shilly-shally. It’s better to be wrong sometimes than to equivocate…”. The readers follow Halmos’s career, and every mathematician who crossed his way (including himself) is given a supposedly fair yet ruthless treatment. This is great book.

Halmos is happy to sort mathematicians into ranks: Gauss and Archimedes are mathematicians of the first rank, Klein and MacLane the second, Mackey, Tarski and Zygmund the third. He puts himself in the fourth rank, together with Birkhoff and Kuratowski. Immediately after discussing ranks, he introduces Fomin, a mathematician who he met in Moscow: “As a mathematician, he was perhaps of rank five”.

The final chapter of the book is called “How to be a mathematician”. I am guessing (a wild guess) that Halmos considered this as a possibility for the title of the book, but realised that it’s the wrong title. A more precise title for the book would be “How to be Halmos”. In the ruthless spirit of the author, one might also suggest: “How to be a great mathematician without really being one”.

Perhaps that’s precisely what makes the book so interesting to me. It is written by an unquestionably human mathematician. Smart, innovative, talented, idiosyncratic, hard working, ambitious – yes, but still human. An important mathematician, but not a Great One. I recommend it, it is fun to read whether or not you agree with what he has to say. I have a lot of criticism on his views, but man does he know how to write!

(Well, the book is perhaps too long and has it’s ups and downs. But one is free to skip the boring “funny” stories on the incompetent waiter in Moscow, or adventures in Uruguay).

I cannot resist objecting loudly to two pieces of advice that Halmos gives.

Halmos writes “…to stay  young, you have to change fields every five years.” Watch out (everyone except Terry, yes?): that is dangerous advice! 

I personally love to branch out and work on different kinds of problems, and to learn things in different fields, but if you are interested in reaching into the deep you have to focus on some concentrated part of mathematics for a long time, for years. I have no regrets, but my experience taught me a few things that one should take into account. When you switch fields the expertise which you acquired becomes pretty much useless and you have to invent or learn new techniques from scratch. To become a reliable scholar in a new area you have to pay an expensive entrance fee by learning the literature, and your investment in the literature of the previous field goes to waste, at least in some sense. From a pragmatic point of view, it will be hard to get good letters for your promotion if you don’t stick long enough in one field to make an impact. And you may receive invitations to workshops and conferences that are no longer very relevant to you, while you are not yet recognised by the people organising workshops that you would like to go to.

It is very hard to be a true expert, a learned scholar, and to make an impact even in one field. Halmos worked on measure theory, ergodic theory, probability, statistics, operator theory, and logic. It is very very unusual, and I don’t know if Halmos is really an exception, for someone who is even very strong to make deep contributions in logic as well as in operator theory. Well, at least in this time and age it is very unusual – remember that Halmos was born 100 years ago, and mathematics has changed since the 40s and 50s quite a lot. But I think that changing fields dramatically and often was bad advice even when Halmos was active. Would his contributions to operator theory been deeper if he had not left it for several years to work on logic?

Of course, if an opportunity to branch out comes along, if your heart pulls you to a different subject, if one problem leads you naturally into a different field, then go for it! But changing fields is not an item on your checklist. Contrary to what Halmos writes, “if a student writes a thesis on the calculus of variations when he is 25, and keeps publishing papers on the calculus of variations till he is 65”, he certainly may be a first rate mathematician.

The second piece of Halmos wisdom I wish to denounce is something that appears in the chapter “How to be a mathematician”, a piece which has appeared separately and which I bumped into already many years ago, and has annoyed me even then.

Halmos writes: “[to be a mathematician] you must love mathematics more than anything else”. He goes on:

To be a mathematician you must love mathematics more than family, religion, money, comfort, pleasure, glory.

What!? More than your children? Well, Halmos did not have any children, and he probably would not have written that line if he did. But even if you don’t have children, really? Do you love mathematics more than love? More than making love? I reject this point of view altogether.

Sure, it’s not just “a job”. You shouldn’t (and couldn’t) be a mathematician if you are not thrilled by it, if it does not captivate your thoughts sometimes to the point of obsession. And you won’t succeed unless you are very devoted, unless you work with joy and work very hard. But if math is more important to you than everything else, then you are simply nuts. It can’t be more important to you more than everything else, because it’s not. In any case, there are many counter examples to the above assertion; many (all?) great mathematicians had loves, devotions, or callings, bigger than mathematics.

In fact, I believe that Halmos himself is a counter example to his claim. You can find the proof in the first and last few paragraphs of the book. These are among the most touching passages in the book, so I will just leave it at that.

 

*****

Apropos Halmos’s book, I take this opportunity to NOT recommend – meaning recommend not to read – Hardy’s book “A Mathematician’s Apology” (Prof. Hardy: apology not accepted!) together with Littlewood’s “A Mathematical Miscellany” (who cares?). I read Hardy’s book because a friend recommended it very highly, and I read Littlewood’s book as a possible compensation, or better: retaliation, for reading Hardy’s book. My verdict: bad books, don’t waste your time with either of these!