Functional analysis – a preface to the introduction

by Orr Shalit

I am planning to write a post that will be an introduction to the course “Advanced Analysis”, which I shall be teaching in the fall term. The introduction is to comprise two main themes: motivation and history. I was a little surprised to find out – as I was preparing the introduction – that, looking from the eyes of a student, the history of subject provided little motivation. I also began to oscillate between two opposite (and equally silly) viewpoints. The first viewpoint is that functional analysis is a big and respectable field of mathematics, which needs no introduction; let us start with the subject matter immediately since there is so much to learn. The second viewpoint is that there is absolutely no point in studying (or teaching) a mathematical theory without understanding its context and roots, or without knowing how it applies to problems outside of the theory’s borders. Pondering these, I found that I had some things to say before the introduction, which may justify the introduction or give it the place I intend.

Last year I taught a course on harmonic analysis for engineers, and in one of my lectures (somewhere mid-term, after we finished altogether with Fourier series) began thus:

Me: Good morning! Today we begin to discuss the Fourier transform. Definition: Let f be a function. Then the Fourier transform of f is defined to be integral-from-minus-infinity-to-plus-infinity-of-f(x)-times-e-to-the-i-omega-x-dx…..

Rude Student: (interrupting me) But where did this come from? What is it good for? Why do we need to learn it?

This type of behaviour typically gets me into a rage. Do the students have no patience? Do they have no faith? They will see applications soon enough, and plenty of them, but now we must first learn the tools. How will be able to build a chair before we know how to use a hammer?  And besides, if it is part of the requirements for an engineering degree, then can’t the students understand that there has to be a good reason for this material to be included, can’t they see that somebody must have thought about it, and that somebody decided that every engineer needs to know this stuff, etc. etc.

Well, that’s very typical of us mathematicians – we are sometimes so deeply immersed in our little puddle that we forget how it looks like from the outside. Indeed, we wonder why anybody would ever want to be outside of this puddle. Oh, let me stop speaking about “us mathematicians” and just speak of myself.

When I was a student, there seemed to be no bound to my patience and my faith. I would swallow anything you would feed me with good faith, and I would patiently digest it. Every new object was exciting, every new category was a new love. Here is the definition of a group, let’s study the consequences of the axioms. Oh, you can also define rings, they have more structure! Oh-oh, you can define modules over rings, that’s two degrees of freedom in abstractness!! If a group has no inverses it is called a monoid, and that’s way more general than a group – good to know that these things have a name, there certainly has to be an interesting theory for this….

As the years went by, more and more objects came, more and more categories went. I myself had made some new definitions which I had no doubt were precisely the concepts that were missing from the subject and would now change it forever. I still think that the objects defined by myself are interesting and must be studied, but looking at the works of others I sometimes get the impression that anyone can define anything and then study it. Anyone can pose any problem and then work on it for years. I have heard at least two of my friends (mathematicians both) ask the following question: “How can this be generalized?”. But, my friends, why  should this be generalized? Should it, really? 

Thus, with time and experience, I started to lose my patience for new definitions (Speaker: quantum group is … Voice in my head: who cares?) My faith gave way to incredulity (Sorry, Book, I do not believe you that the theory of C*-correspondences really has applications in physics, and twenty references at your end will not convince me otherwise). It would really be fantastic if a majority of problems one can pose would be worth pursuing, but I have come to think, at times, that even the majority of problems actually posed might not be worth pursuing.

Well, if so, then what problems are worth pursuing? What theories are worth developing?

My goal in this post is not to answer that. Nor is my goal to undermine pure or abstract mathematics. My goal is to explain why I think “introductions” are so important, and that mathematicians should never lose their broad view, not only of their subject but far beyond their subject. To this I would like add that I think that my rude student had a point (no doubt, all his questions have very good answers in the context of the Fourier transform). The point is that one should carry a certain question in ones mind no matter what one sets out to do: Is this a worthwhile endeavor, and why is it so? 

The next post will be an introduction to the course “Advanced Analysis”, which is a course in functional analysis. I wish to close this post by quoting Vladimir Arnold:

In the last 30 years, the prestige of mathematics has
declined in all countries. I think that mathematicians
are partially to be blamed as well (foremost, Hilbert
and Bourbaki), the ones who proclaimed that the goal
of their science was investigation of all corollaries of
arbitrary systems of axioms.

(V. Arnold, in 1990 interview to the magazine “Kvant”, English translation taken from “A Tribute to Vladimir Arnold”, Notices of the American Mathematical Monthly, link).

Update, 20.9.2012: In my defense (or in defense of quantum groups), let me add that eventually I heard a talk on quantum groups by Adam Skalski which was highly motivated and interesting and in that talk there were no voices in my head. That was just to prove a point, you see.