Noncommutative Analysis

Category: Books

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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A list of corrections to my book

My friend Daniel Reem has encouraged me to put up a webpage with corrections to my book “A First Course in Functional Analysis” (in fact, he even convinced me to insert a promise to put up such a page in the preface to my book, so that I don’t really have an option not to put up that corrigendum). Daniel was also kind enough to find a couple of mistakes! So thanks to Daniel, the page is up: here is the link.

Anyone who wishes to report mistakes can use the comment section in this page, or email me.

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 L^2[a,b] as a completion of C([a,b]), 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 L^2[a,b] as a completion space is not new, few (if any) have attempted to actually walk the extra mile and work with L^2 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 integral 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.

The preface to “A First Course in Functional Analysis”

I am not yet done being excited about my new book, A First Course in Functional Analysis. I will use my blog to advertise my book, one last time. This post is for all the people who might wonder: “why did you think that anybody needs a new book on functional analysis?” Good question! The answer is contained in the preface to the book, which is pasted below the fold.

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