Lebesgue integration well predates quantum field theory, and it’s pretty clear that the Feynman path integral is not the same kind of integration as Lebesgue. I’m not sure the exact mathematical status of the Feynman path integral, but it’s conventional wisdom among physicists that the correct formulation of quantum mechanical systems (including quantum field theories) is as a unitary operator on a Hilbert space.

The question about whether X exists linear maps from X to itself is the sum of a compact operator and a scalar multiple of the identity is interesting because it might be false. We know precisely two ways to construct continuous linear maps on an arbitrary Banach space: we can construct compact operators, and we can construct scalar multiples of the identity. Is there a third way that works for arbitrary Banach spaces? I’m sure experts think the answer is no, but it’s an important unresolved question.

hellblazer said: “I don’t really follow the first and second sentences of this claim…”
What i was trying to say is that the subjects number theory, combinatorics, geometry and physics are in way closer to the origin of mathematics than subjects like category theory or Banach spaces. The latter two evolved in a long process out of problems in the first few. You don’t sit down one bored afternoon and out of the blue ask yourself “is there an infinite dimensional Banach space X such that every bounded linear map from X to itself is the sum of a compact operator and a scalar multiple of the identity”. While asking yourself if there is an infinite number of twin primes is more likely to happen. (btw i would really be interested in hearing why the first question is of importance to experts)

I’m not asking for mathematics to be geared toward end product, but on the other hand the possibilities of writing down some axioms and asking all possible questions about them are endless. And unfortunately the modern way of teaching mathematics conveys somewhat this spirit to the student. Which ultimately also influences on the way research in mathmatics is done nowadays.

mike said:
What concerns number theory and combinatorics i think of them differently than e.g. abstract algebra. These are subjects where questions originate as is geometry or physics. While subjects like point set topology, homological algebra, functional analysis etc. were discovered while trying to deal with such questions. These “secondary” subjects should be studied in their own right, but without forgetting their place in the bigger picture.

I don’t really follow the first and second sentences of this claim, since I fail to see how, say, the twin prime conjecture or the classical results on quadratic number fields with class number 1, or the Hales-Jewett theorem or Rota’s beloved algebraic approach to Mobius inversion, are motivated by questions “originating in geometry or physics”. Perhaps I’m just missing the Bigger Picture.

That was me trying to be polite, btw. So let’s get on to the last sentence of Mike’s dictum, that these “secondary” subjects can be studied in their own right (how gracious of you) but without forgetting their place in the bigger picture. This seems to be based on a more general contention that academic education and research should be geared toward “end product”, and while I don’t have adequate knowledge or space to rant about this I would just like to voice my strong disagreement with this sentiment — if it is what’s being implied. Apologies if not.

Finally: functional analysis was indeed motivated by PDE and quantum theory (not QFT itself AFAIK) but to say that
as soon as you leave the abstract generalities and try to apply it to problems in PDE’s or QFT (which as i think was the original motivation for developing the subject) you start to run into difficulties that become very technical and hard fast.
does not mean that it is not useful — it just means that the original ares, namely PDEs and QFT, are *very hard*. If mike has thoughts on the *right way* to do these which does not involve functional analysis then many people would love to know! I mean, without the functional analytic framework how does one study densely defined closed unbounded operators (rigorous solution of Schroediinger equation)? Or distribution theory for PDEs? or stochastic calculus, if one wants more “real-world” applications?

Sorry if this is an over-agressive reaction, it’s been a tiring day and being told that something you’ve studied should be subservient to the Big Picture is, for some of us, somewhat trying.

Secondly, I just don’t agree that functional analysis isn’t applicable. As I said, I think you really need it to fully understand some of the issues in topological group theory, even nicely behaved stuff like compact Lie Groups. I’m also often surprised at how much time books in this area spend over functional analytic issues: I suspect this is because they fear that their audience isn’t that sophisticated in such issues.

I’ve recently become pretty interested in compact quantum groups, a la Woronowicz, and while it’s possible to setup a formulation which doesn’t use much functional analysis, it’s interesting that Woronowicz’s own thinking seems to have drifted, over time, to the C*-theoretic formulation, which I certainly think is more elegant, and more powerful, than the other more ad hoc setups. Indeed, I’m actually a bit fan, al van Daele, Vaes, Kustermanns et al of the von Neumann setting for locally compact quantum groups. Certainly the study of quantum groups came out of physics, and AFAIK still has strong links there.

Finally, I disagree that functional analysis only came out of physics and PDEs etc. I think functional analysis is a natural out growth from real analysis. For example, the Lebesgue integral is surely the “correct” way to view integration, and it’s best explored in the setting of Banach and Hilbert spaces. I’m thinking here of Runde’s books, for example.

I don’t oppose teaching pure math like abstract algebra, since the matters tought in an abstract algebra course appear almost everywhere once you start to see the bigger picture. But there is a danger that some people, before they arrive to see the bigger picture, start to believe that e.g abstract algebra is just sudied for its own sake. And they start to work in that area asking questions of a purely abstract and axiomatic nature, which are of almost no importance to the original problems. I consider this sort of mathematics somewhat a waste of energy. And i think some parts of functional analysis are like that.

What concerns number theory and combinatorics i think of them differently than e.g. abstract algebra. These are subjects where questions originate as is geometry or physics. While subjects like point set topology, homological algebra, functional analysis etc. were discovered while trying to deal with such questions. These “secondary” subjects should be studied in their own right, but without forgetting their place in the bigger picture.

I suppose you could throw out the complex numbers entirely and get “p-adic Banach spaces”…

I am just a beginner, I don’t believe I have even heard the claim that the reals and the the complexs are the only complete subfields.