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.

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.

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 …

> 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.)

Sobolev spaces, Banach spaces, Hardy spaces, Hilbert spaces, Besov spaces, Frechet spaces and countless others.

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.

If you like reverse engineering then you don’t need to stop at point set topology. Reverse Mathematics is a curious field.

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”?

>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 )

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!