### 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:** *A **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.

A few remarks:

0. Good luck in the new blog!

1. I remember that you told me the story with the rude student.

2. In many cases it is not clear in advance why a certain theory

is worth to be studied/developed beyond the fact that it is interesting

for the ones who do the job. Sometimes one can start in one place, very

abstract and seems to be related to nowhere, and finish in a completely

other place, very practical one. This happened to me personally but of

course there are many well known and good examples in the history of

math.

Example: you mentioned group theory. Well, this theory, and in particular the

almost forgotten theory of Lie groups and Lie algebras, has found many

applications in physics (e.g., nuclear physics and quantum mechanics)

many years after it was studied by pure mathematicians.

I don’t claim that any theory or a mathematical paper is good: there are

many ones which are artificial and are not even beautiful. But still, it is not

clear in advance what will prove to be useful (if this the goal). Sometimes

only after many years a theory reaches a maturity and beauty. Sometimes

the practical importance of a theory is merely because it gave some

inspiration to other theories, or helped to clarify certain notions or certain

method of thinking and so on. And another fact that one should take into

account is that the majority of works and theorems are doomed to be

forgotten, even very good and very beautiful ones. That’s a basic fact which

can be easily checked by looking at the citation index of the existing papers.

Only very few of them have more than a few citations and are known to

more than a few people. And even the well known papers may have parts

which have completely forgotten: I myself had the honor to save from the

dark of oblivion a very beautiful and useful result (at least to me)

which was almost totally forgotten (I told you and other about it). And

besides, who says that the popular theories are the good or important ones

(unless the measure of importance of a theory is its popularity)?

Math is a kind of a creative activity and at least one can have certain happy moments whenever one discovers/proves something interesting. I am not sure that this is less worth than watching TV or playing basketball.

3. An interesting paper about the history of the Hahn-Banach theorem and other aspects of functional analysis is:

Lawrence Narici and Edward Beckenstein, “The Hahn–Banach Theorem: The Life and Times”, Topology and its Applications, Volume 77,

Issue 2 (1997), 193–211

http://at.yorku.ca/p/a/a/a/16.pdf

4. A minor remark: the paper about Vladimir Arnold can be found below.

The title and the journal a slightly different.

http://www.ams.org/notices/201203/rtx120300378p.pdf

Thanks for the comment and the links! I’ve added your link to the tribute to Arnold near the quote.

1. A few small corrections to my previous message:

A. Item 2: “save from the dark of oblivion” ==>

save from the darkness of oblivion

B. Item 4:

“The title and the journal a slightly different” ==>

The title and the journal are slightly different.

2. A small remark to you: don’t forget to read again (the corrected)

Item 4 since it contains a certain information regarding the

non-existence of a certain paper and a certain journal.

3. Another remark: is it possible to make the links to appear in

blue? at the current configuration it is not completely easy to

distinguish between links and regular text.

Do you mean the catastrophe thing? It’s a good one. Some people make it seem like mathematics is nothing but a collection of missing citations and bad references. Take into account, though, that I have read this charge rebutted.