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I'm a mathematics student in abstract algebra and algebraic geometry. Most of my books cover a great deal of category theory and it is an essential tool in understanding these two subjects.

Recently, I started taking some functional analysis courses and I discovered that there is almost no category theory done in these courses. But since most of the spaces studied in functional analysis are objects in categories (e.g. the normed spaces form a category), I find it rather strange that the books leave the category theory out.

Is there a reason for this?

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I suppose there are a lot of reasons they don't use category theory, and hence don't need to cover it. Do you see some areas of analysis where it would be of use? –  Thomas Andrews Jul 10 '12 at 21:05
I don't have a good answer to your question, but this MO question might be of interest to you: mathoverflow.net/questions/38752/… –  Dane Jul 10 '12 at 21:13
You may wish to consult Helemskii's Lectures and Exercises on Functional Analysis. –  t.b. Jul 10 '12 at 21:16
Another somewhat related MO question: mathoverflow.net/questions/22359/… –  Jonas Meyer Jul 10 '12 at 21:43
Hummm, and here I had always thought analysts did quite a bit with category notions :) –  Dave L. Renfro Jul 10 '12 at 21:48
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6 Answers

Let me ask you a dual question: I am a mathematics student in set theory, why don't category theory students do set theory? I find it strange that most books on category theory have only a naive handling of set theory.

Now let me answer your question. Category theory is an impressive tool for abstraction, but analysis is not always in need for abstraction - it looks for concrete solutions and ideas. In that aspect categories are not too useful. In fact, if you try to insist on concreteness, categories can become a burden when you insist to carry the category around instead of just talking about functions and spaces.

On the other hand, abstract algebra is very fitting for category theory. We focus on "all" groups or "all" modules over a certain ring. Analysis focuses on particular spaces, continuous/differentiable/analytical functions over $\mathbb C$, for example. The situation is similar with set theory, while a very useful tool for formalizing arguments in the better parts of mathematics - it is often neglected and cast aside as a non-issue.

The best tip to remember (as a set theory student) is that while a screwdriver is a very useful tool to carry, you don't really need it if you are making a sandwich.

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Your choice of dual question is interesting. I got interested in category theory precisely because it does do set theory: specifically, the way in which Topos theory provides a more algebraic version of set theory. –  Hurkyl Jul 10 '12 at 21:54
Asaf, you might enjoy One Good Turn by Witold Rybczynski.en.wikipedia.org/wiki/One_Good_Turn_%28book%29 or books.google.com/books/about/One_Good_Turn.html?id=nv9L_FxyhuIC I'm looking at it now, also a few pages on early guns. I have been wondering since these are used in battle in an American cable TV series called The Borgias. Called an arquebus in English or French, a sort of small cannon with a stick for support. Dangerous for the user and intended victims. See: sho.com/sho/the-borgias/home The actors are mostly British, maybe not really American series. –  Will Jagy Jul 10 '12 at 22:33
@Asaf : Does it depend on the kind of sandwich? Perhaps that there exists some sandwich in which case a screwdriver is really relevant to have in hand. (In other words, nice analogy.) –  Patrick Da Silva Jul 11 '12 at 0:02
Any time I hear set theory or (less likely) category theory, screw drivers and sandwiches are gonna pop to mind. –  hydroparadise Jul 11 '12 at 3:49
I wonder if whoever downvoted this is one of those people making sandwiches with screwdrivers... :-) –  Asaf Karagila Jul 11 '12 at 5:53
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There are two questions here, in reality, I think.

First, in brief, I am told by many people that I "do functional analysis in the theory of automorphic forms", and I certainly do find a categorical viewpoint very useful. Second, in brief, it is my impression that the personality-types of many people who'd style themselves "(functional) analysts" might be hostile to or disinterested in the worldview of any part of (even "naive") category theory.

In more detail: as a hugely important example, I think the topology on test functions on $\mathbb R^n$ is incomprehensible without realizing that it is a (directed) colimit (direct limit). The archetype of incomprehensible/unmotivated "definition" (rather than categorical characterization) is in Rudin's (quite admirable in many ways, don't misunderstand me!) "Functional Analysis"' definition of that topology.

That is, respectfully disagreeing with some other answers, I do not think the specific-ness of concrete function spaces reduces the utility of a (naive-) categorical viewpoint.

From a sociological or psychological viewpoint, which I suspect is often dominant, it is not hard to understand that many people have a distaste for the structuralism of (even "naive", in the sense of "naive set theory") category theory. And, indeed, enthusiasm does often lead to excess. :)

I might claim that we are in a historical epoch in which the scandals of late 19th and early 20th century set theory prey on our minds (not to mention the mid-19th century scandals in analysis), while some still react to the arguable excesses of (the otherwise good impulses of) Bourbaki, react to certain exuberances of category theory advocates... and haven't yet reached the reasonable equilibrium that prosaically, calmly, recognizes the utilities of all these things.

Edit: since this question has resurfaced... in practical terms, as in L. Schwartz' Kernel Theorem in the simplest case of functions on products of circles, the strong topology on duals of Levi-Sobolev spaces is the colimit of Hilbert space topologies on duals (negatively-indexed Levi-Sobolev spaces) of (positively-indexed) Levi-Sobolev spaces. As I have remarked in quite a few other places, it was and is greatly reassuring to me that a "naive-categorical" viewpoint immediately shows that there is a unique (up to unique isomorphism) reasonable (!) topology there...

Similarly, for pseudo-differential operators, and other "modern" ideas, it is very useful to recast their description in "naive-categorical" terms, thereby seeing that the perhaps-seeming-whimsy in various "definitions" is not at all whimsical, but is inevitable.

A different example is characterization of "weak/Gelfand-Pettis" integrals: only "in my later years" have I appreciated the unicity of characterization, as opposed to "construction" (as in a Riemann/Bochner integral).

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Very interesting. Do you have any reference you'd suggest for the definition of the topology on test functions from the categorical viewpoint? –  Malik Younsi Jul 11 '12 at 1:09
@MalikYounsi: I recommend looking at the many wonderful notes at prof. Garrett's website; I myself have benefited from them several times. –  wildildildlife Jul 11 '12 at 15:17
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I would guess that a lot of what analysis studies is about individual "points" in a way that isn't suitable for category theory. You are often interested in functions that might or might not be in some collection of functions, and it is not always clear that composition of these functions makes any sense. You are often only interested in showing properties at individual points in a space - it is continuous at $x$ or differentiable or bounded. Category theory is less useful for that sort of problem.

Another area where you might even more expect category theory to be involved is linear algebra. In linear algebra, we are entirely talking about composable functions, etc. But the reality is, while category theory gives you a nice overview way of thinking of linear algebra, it doesn't give you any real aid in solving problems about linear algebra.

The most powerful things we can do with category theory are when we start related seemingly different categories. That shows up in analysis in some ways, but at the lower level, the examples are more easily understood as linear algebra (for example, Fourier series.)

It's not that there isn't some useful ways to use category theory in analysis. I'm not well-versed enough to know where it is used or not. However, it seems to me that understanding the fundamentals is not aided enough by category theory to make it worthwhile.

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I disagree with the second paragraph. I don't know any reasonable explanation of the difference between a vector and the components of a vector (w.r.t. change of basis for example) which doesn't ultimately point out the difference between a covariant and a contravariant functor. Similarly I think determinants and exterior products are cleanest to develop functorially (and you really do use the functoriality: it is a coordinate-free version of looking at minors). –  Qiaochu Yuan Jul 10 '12 at 21:25
How does that explanation help you solve any sort of linear algebra problem, @QiaochuYuan. It clarifies the concept and puts it in a broader context, but it doesn't really help solve problems. –  Thomas Andrews Jul 10 '12 at 21:28
Tell that to my younger self when I was taking a class on special relativity... I was constantly getting confused about the difference between a vector in Minkowski space and its components and looking back I strongly suspect I did a lot of problems incorrectly because I applied Lorentz tranformations in the wrong direction. –  Qiaochu Yuan Jul 10 '12 at 21:29
In other words, understanding linear algebra functorially helps you figure out whether you want to multiply your vectors by the change of basis matrix or by the inverse of the change of basis matrix or maybe by its transpose... I think it is useful in solving problems not to be confused about such things! –  Qiaochu Yuan Jul 10 '12 at 21:32
@QiaochuYuan Ah, from my point of view, I thought of the difference as much like the difference between a number in base 10 and a number in base 7. The same number, different "expression." But I agree that quite a bit can be revealed by looking at linear algebra from a category-theoretic point of view, I just don't think it is pedagogically appropriate way to approach the topic. –  Thomas Andrews Jul 10 '12 at 21:32
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If you want to study functional analysis from categorial point of view see this book. Its first chapter starts from introduction to category theory.

I think that this approach is rare because most functional analyst study very concrete objects and spaces. So category theory become useless for them.

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You might also Take a Look at the Book "the geometry of PDE's and mechanics" by agostino Prastaro. You can find a google books version online. For someone looking for category theory in analysis this ought to be fun.. Especially check out the parts about spectral sequences. :)

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The level of abstraction inherent in category theory makes it harder to apply it to some of the very concrete questions analysts look at. If you are trying to find Sobolev space properties of solutions to the heat equation just to give an example, the methods currently available to address such questions are pretty hands-on. Fourier transforms, conserved quantities, pseudodifferential operators, etc. The same applies in Fourier analysis in ${\mathbb R}^n$ and various other "concrete" subjects.

There are other areas of analysis where categories are relevant, although somewhat indirectly. In several complex variables for example, they use sheaves and cohomology and so on. I believe most several complex variables people don't use categories that much, but there are analogues of what they do in parts of algebraic geometry where they do use category theory.

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