Noncommutative Analysis

Category: Book Review

Tapioca on page 49

To my long camping vacation this year I took the book “Topological Vector Spaces” by Alex and Wendy Robertson. I “inherited” this book (together with a bunch of other classics) from an old friend after he officially decided to leave academic mathematics and go into high-tech. The book is a small and thin hard-cover, with pages of high quality that are starting to become a delicious cream color.

I decided to read this book primarily because I like to read the books I have, but also because I am teaching graduate functional analysis in the coming semester and I wanted to amuse myself by toying with the possibility of de-emhasizing Banach spaces and giving a more general treatment that includes topological vector spaces. I enjoyed thinking about whether it can and/or should be done (the answers are yes and no, respectively).

Oh sister! I was pleasantly surprised with how much I enjoyed this book. They don’t write books like that any more. Published in 1964, the authors follow quite closely the tradition of Bourbaki. Not too closely, thankfully. For example they restrict attention from the outset to spaces over the real or complex numbers, and don’t torture the reader with topological division rings; moreover, the book is only 158 pages long. However, it is definitely written under the influence of Bourbaki. That is, they develop the whole theory from scratch in a self-contained, clean, efficient and completely rigorous way, working their way from the most general spaces to more special cases of spaces. Notions are given at the precise place where they become needed, and all the definitions are very economical. It is clear that every definition, lemma, theorem and proof were formulated after much thought had been given as to how they would be most useful later on. Examples (of “concrete” spaces to which the theory applies) are only given at the end of the chapters, in so called “supplements”. The book is rather dry, but it is a very subtly tasty kind of dry. The superb organization is manifested in the fact that the proofs are short, almost all of them are shorter than two (short) paragraphs, and only on rare occasion is a proof longer than a (small) page. There is hardly any trumpet blowing (such as “we now come to an important theorem”) and no storytelling, no opinions and no historical notes, not to mention references, outside the supplement. The author never address the reader. It seems that there is not one superfluous word in the text. Oh, well, perhaps there is one superfluous word.

After the definition of a precompact set in a (locally convex) topological vector space, the authors decided to illustrate the concept and added the sentence “Tapioca would make a suitable mental image”. This happens on page 49, and is the first and last attempt made by the authors to suggest a mental image, or any other kind of literary device. It is a little strange that in this bare desert of topological vector spaces, one should happen upon a lonely tapioca, just one time…

* * * * *

So, why don’t people write books like that any more? Of course, because this manner of writing went out of style. It had to become unfashionable, first of all, simply because old things always do. But we should also remember that mathematical style of writing is not disconnected from the cultural and philosophical surroundings. So perhaps in the 1930s and up to 1950s people could write dogmatically and religiously about mathematics, but as time went by it was becoming harder to write like this about anything.

In addition to this, it is interesting that there were also some opposition to Bourbaki, from the not much after the project took off, and until many many years later.

Not that I myself am a big fan. I personally believe that maximal generality is not conducive for learning, and I prefer, say, Discussion-SpecialCase-Definition-Example-Theorem-Proof to Definition-Theorem-Proof any day. I also don’t believe in teaching notions from the most general to the more specific. For example, in my opinion, set theory should not be taught-before-everything-else, etc. For another example, when I teach undergraduate functional analysis I start with Hilbert spaces and then do Banach spaces, which is inefficient from a purely logical point of view. But this is how humans learn: first we gurgle, then we utter words, then we speak; only much later do we learn about the notion of a language.

So, yes, I do find the books by Bourbaki hard to use (reading about all the pranks related to the Bourbaki gang, one cannot sometimes help but wonder wether it is all a gigantic prank). But I have a great admiration and respect for the ideals that group set and for some of its influences on mathematical culture. The book by Robertson and Robertson is an example of how to take the Bourbaki spirit and make something beautiful out of it. And because of my admiration and respect for this heritage, it is a little sad to know that Bourbaki was quite violently abused and denounced.

If you have ever read some harsh and mean criticism of the Bourbaki culture, if you have heard someone try to insult someone else by comparing them to Bourbaki, then please keep in mind this. Nobody really teaches three-year-olds set theory before numbers. In the beginning of every Bourbaki book (“To the reader”), it is explicitly stated that, even though in principle the text requires no previous mathematical knowledge on the part of the reader (besides the previous books in the series) “it is directed especially to those who have a good knowledge of at least the content of the first year or two of a university mathematics course”.  Bourbaki didn’t “destroy French mathematics” or any other nonsense. The source of violent opposition is not theological or pedagogical, but psychological. In my experience, the most fervent opponents of the Bourbaki tradition who I heard of, are people of non-neglible egos (and their students), who were simply very insulted to find out that a self-appointed, French-speaking(!) elite group decided to take the lead, without asking permission or inviting them (or their teachers). That hurt, and a crusade, spanning decades, ensued.

* * * * *

Well, let us return to the pleasant Robertsons. Besides the lonely tapioca, I found one other curious thing about this book. On the first page the names of the authors are written:

A.P. Robertson

(Professor of Mathematics

University of Keele)

AND

Wendy Robertson

So, what’s the deal with A.P. and Wendy? Is A.P. a man? I guessed so. Are they brother and sister? Why is he a professor and she isn’t? Are they father and daughter? I wanted to find out. I found their obituaries: Wendy Robertson (she passed away last year) and Alexander Robertson.

So they were husband and wife, and it seems that they had a beautiful family and a happy life together, many years after writing this book together. I remained curious about one thing: whose idea was it to suggest tapioca? Did they immediately agree about this, or did they argue for weeks? Was it a lapse? Was it a conscious lapse?

* * * * *

In the course that I will teach in the coming semester, I am not going to use the language of topological vector spaces. I will concentrate on Banach spaces, then weak and weak-* topologies will enter. These are, of course, topological vector spaces, but there is no need to set up the whole framework to notice this, and there is no need to prove everything in the most general setting. For example, the students will be able to prove a Hahn-Banach extension theorem for, say, weak-* continuous functionals, by imitating the proof that I will give in class in a similar setting.

On Saturday I went to my nephew’s Bar-Mitzva, and they had tapioca for desert (not bad), and I thought about Wendy and Alex Robertson. Well, especially about Wendy. I think that it was her idea.

 

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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!