Against Functional Analysis
April 6th, 2007 by WaltThe functional analysis thread has turned into a big love-fest, but a dissenting voice has appeared from an unexpected quarter. David Corfield spotted this interview with Gian-Carlo Rota and David Sharp. Rota says:
Combinatorics is an honest subject. No adèles, no sigma-algebras. You count balls in a box, and you either have the right number or you haven’t. You get the feeling that the result you have discovered is forever, because it’s concrete. Other branches of mathematics are not so clear-cut. Functional analysis of infinite-dimensional spaces is never fully convincing; you don’t get a feeling of having done an honest day’s work. Don’t get the wrong idea-combinatorics is not just putting balls into boxes. Counting finite sets can be a highbrow undertaking, with sophisticated techniques.
This raises a disturbing possibility: I like functional analysis because it provides yet another way to avoid an honest day’s work.
April 6th, 2007 at 9:13 pm
I think the analogy is silly. If you’re going to say that combinatorics is about putting balls into boxes, then you might as well say that analysis only looks at differentiable functions. Of course, both of them are nonsense. A lot of the stuff in volume 2 (especially) of Stanley’s Enumerative Combinatorics is (at least to me) extremely abstract. I don’t find noncommutative generating functions any more concrete than infinite-dimensional Hilbert spaces.
April 6th, 2007 at 9:58 pm
Well if you followed the link to Corfield’s post and read the it you’d see John Baez note that
And — as he notes — if you followed the link to the article you’d see Rota himself say that
Reading quotes in context is almost as fun as Walt thinks functional analysis is.
April 7th, 2007 at 4:10 am
This seems a little ironic to me, as at the far, abstract end of functional analysis (e.g. certain bits of Banach space theory) gets very, very close to issues in set theory and combinatorics. For example, a lot of Gowers’s work is cast in the language of infinite games. Furthermore, a lot of Gowers’s work in combinatorics uses notions from functional analysis, although usually applied in finite cases (e.g. Fourier transforms over finite fields). AFAIK this equally applies to work of Green and Tao. At least here in the UK, there is a pretty big cross-over between people interested in functional analysis and people interested in combinatorics (Gowers and Bollobas come to mind immediately). Finally, I get the vague impression (warning: I’m a functional analysist, and paranoid) that other mathematicians view both fields with a bit of scorn!
April 7th, 2007 at 5:17 am
John: The Rota quote about sophisticated technique in your comment is already in my post.
April 7th, 2007 at 7:30 am
Doormat wrote:
True, but don’t mathematicians in every field view all other fields with a bit of scorn?
This could make a fun topic of conversation, Walt - especially if you’re trying to avoid big love-fests. Which branches of mathematics view which other branches with scorn?
It’s a time-dependent business. In the 1950s I get the feeling that highly abstract algebraic geometers and algebraic topologists were regarded, at least by themselves, as the most noble of mathematicians. Bourbaki certainly regarded themselves as the kings of mathematics.
I imagine these high-falutin’ fellows would regard combinatorists counting balls in boxes with a bit of scorn, saying the field was “just a bunch of tricks”. Later Rota came along and fought this impression quite mightily by pushing general techniques like Hopf algebras, generating functions, the umbral calculus, and Richard Stanley.
Functional analysis saw its heyday somewhat earlier - it was, I believe, once quite as prestigious as algebraic geometry was in the 1950s, but it somehow lost its clout after solving a lot of its most pressing problems.
By the 1980s, the same thing had happened to algebraic topology - or at least “classical” algebraic topology, centered on homotopy theory. When I entered MIT as a grad student in 1982, I remember being told that “homotopy theory is dead”. Presumably this meant that the remaining open problems were so abstract and/or difficult that few people could work on them. The word on the street was that the exciting part of algebraic topology was now its interactions with differential topology and analysis. The Atiyah-Singer index theorem! Everyone wanted to prove new versions of this, and find shorter proofs… if you didn’t want to do that, you were square.
Now, 20 years later, homotopy theory is back in fashion: turns out it wasn’t dead after all!
Etcetera, etcetera. With all these shifting fashions (and I’m sure they’re different at different places), different groups of mathematicians get to have the pleasure of looking down on other groups with scorn.
April 7th, 2007 at 8:34 am
Walt: /me facepalms
John: A data point. (Some) algebraic geometers view terms from knot theory with a bit of scorn. Witness a junior faculty observing a lecture intended for graduate students on the basics of Khovanov holomogy: “Isn’t it fair to say that the Jones polynomial’s importance has been highly exaggerated?”
April 7th, 2007 at 8:53 am
Functional analysis is a load of old crap. It’s a subject that’s in dire need of a decent mathematician to come along and sort it out. In no other branch of mathematics do you have such ad hoc constructions one after another: Sobolev spaces, Banach spaces, Hardy spaces, Hilbert spaces, Besov spaces, Frechet spaces and countless others. Oh dear, my sequence doesn’t converge! I know, I’ll just rig some kind of space where these sequences are excluded and now my theorem holds. Yippee! Never mind that my theorem was so hideous it didn’t deserve to live, I can still publish a paper on it.
The really depressing thing is knowing that part of my knowledge of Riemann surfaces has a dependency on a form of Serre duality and the only proof I know uses functional analysis. One day I’ll know the algebraic proof and I’ll be able to excise that last bit of functional analysis from my brain.
April 7th, 2007 at 2:51 pm
sigfpe: In no other branch of mathematics do you have such ad hoc constructions one after another
them’s fighting words, friend….
In any case I spend most of my time thinking about $\ell^1$, which I don’t think anyone can claim is an ad hoc creature. Look! it’s even a left adjoint! It must be canonical!
Seriously, though, can people suggest other areas of maths that have tolerated a proliferation of bits-and-pieces?
Never mind that my theorem was so hideous it didn’t deserve to live, I can still publish a paper on it.
Damn! I’ve been rumbled!
April 8th, 2007 at 2:28 pm
hellblazer,
>Seriously, though, can people suggest other areas of maths that have tolerated a proliferation of bits-and-pieces?
Maybe this is what David Corfield means when he talks of Moufang loops and quasigroups not forming part of a good story here. (Sorry, I should rephrase that: he hypothesises the existence of a person who might suggest that
)
April 8th, 2007 at 6:22 pm
“Seriously, though, can people suggest other areas of maths that have tolerated a proliferation of bits-and-pieces?”
Point set topology. But I actually like the flair of these fields. The reverse engeneering approach of “What kind of assumptions do I need for this property to hold”?
April 9th, 2007 at 8:59 am
Michael,
If you like reverse engineering then you don’t need to stop at point set topology. Reverse Mathematics is a curious field.
April 9th, 2007 at 12:48 pm
sigfpe wrote:
That seems mightily unfair. First of all, it seems utterly sensible to study complete inner product spaces (Hilbert spaces) as a special case of complete normed spaces (Banach spaces) as a special case of complete spaces defined by a bunch of seminorms (Frechet spaces). These are all things that show up in real life: you’ve got your square-integrable functions on the circle (a nice Hilbert space), your continuous functions on the circle (a nice Banach space), and your smooth functions on the circle (a nice Frechet space) - we need to understand all of them, and we want general theorems that still apply when someone suddenly replaces the circle by, say, a sphere!
Sobolev spaces are just an example of Hilbert spaces, namely Hilbert spaces where we take functions on some manifold and demand that a certain number of derivatives be square-integrable. You couldn’t get far studying partial differential equations without these! And you can’t blame functional analysts for knowing examples of their general concepts - that’s a good thing.
Similarly, Hardy spaces are just an example of Banach spaces, namely Banach spaces of holomorphic functions on the unit disk such that the p power of their boundary values are integrable. People need these in complex analysis.
Besov spaces show up when you look at boundary values of functions that lie in Sobolev spaces. As a spinoff of a natural concept, these are again unavoidable - but only if you’re studying partial differential equations with boundary conditions. If you don’t care about those, ignore Besov spaces!
Anyway, while the original comment was perhaps made in half-jest, I seriously disagree with it. Complaining that functional analysts have names for various sorts of topological vector spaces is sort of like complaining that algebraists have names for various sort of commutative rings. Personally I shudder each time I hear about “Cohen-Macaulay rings”, “Gorenstein rings”, “Prüfer rings” and the like - but I know that’s just because of my ignorance, which I mean to cure someday. I don’t think it’s some defect of the subject of commutative algebra.
April 9th, 2007 at 3:13 pm
jb,
> Personally I shudder each time I hear about…
My rant was me shuddering loudly
Seriously though, I’d really love to be able to appreciate and be good at functional analysis. Mainly because of the role it plays in quantum mechanics and quantum field theory. But I find it such a painfully dry subject to read. Many other branches of mathematics seem to have champions that really motivate the subject well. (And I’d list you among those people.) But I’ve never found anyone whose writing could make functional analysis seem quite as exciting as I find many other branches of mathematics. (One exception: Narici and Beckenstein talk very excitedly about Hahn-Banach.)
April 9th, 2007 at 4:16 pm
Okay, I admit this is partly an attempt to publicize myself, but there’s some nifty applications of functional analysis hidden in the details of the results quoted in my exegesis of Gregg Zuckerman’s first lecture about real reductive algebraic groups. Even in straight-up algebra there’s a lot to be said for a dash of functional analysis to make things work nicely.
April 10th, 2007 at 10:43 am
Admittedly this is usually classed as Fourier analysis rather than functional analysis, but I’ve always found the Poisson summation formula and Parseval’s identity like unexpected magic tricks. The Hilbert space proof of Radon-Nikodym is also IMHO a nice advert for functional-analytic methods.
I wouldn’t claim to find the Hahn-Banach theorem exciting. Fascinating, maybe, but it doesn’t really put a spring in my step or quicken the pulse …
April 14th, 2007 at 5:25 am
sigfpe: I want to type up your first comment, print it, and hang it on my office door. After adding, in red pen, that _I_ know an algebraic proof of Serre duality!
About your second comment: as a student I enjoyed Rudin’s book. Most of my fellow students were really excited by it, and grew up to be functional analysts.
John Baez: I never understood either Cohen-Macaulay or Gorenstein until I learned the geometric significance. It just means that the dualizing complex has only one nonzero cohomology module (CM) and that one module is locally free of rank one (G). Yes, I am a strange person.
My favorite short reference for both CM and G is at the beginning of Miles Reid’s Young Persons’ Guide to canonical singularities.
April 14th, 2007 at 11:11 am
I wish to comment on some points raised in this and the previous thread regarding functional analysis.
I suspect that the person who stated that functional analysis did not solve any problems in PDE’s is not a very big specialist in PDE’s. Here are just two tiny examples:
a) The spectrum of the Laplacian (in appropriate domain) is a (discrete) sequence converging to infinity (or minus infinity, depends how you define it) - you get that almost for free if you use functional analysis.
b) If a PDE is uniquely solvable than it is also well-posed (”stable”) - immediate from the open mapping theorem.
Functional analysis gives a powerful framework to handle infinite dimensional spaces in a unified way. True, when one applies it, say, to differential equations, one has to decide in what space of functions to work in. The choice, in many cases, is different from the classical spaces of k times continuously differentiable functions. But some applied mathemticians will argue that the spaces of weakly differential functions are more suitable not only for analysis but also for applications (where discontinuous slutions may be natural), and using functional analysis has helped to choose these spaces.
Here is what functional analysis does not give you: it does not remove the need to carry out some hard analytical work. To me this first came as a dissapointment. Later I realized that removing the analysis from analysis altogether is impossible. So, don’t worry, functional analysis has made analysis a little easier, it has put it into a broader context, but it didn’t take the fun out of it.