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Let me start by saying that I don't think this question is "not acceptable", despite the choice of discussion category. However, I have misgivings about it, and to some extent some other questions, because in my view the scope is too broad and rather subjective, and invites people to contribute answers that are more trendy than they are deliberated. Moreover, the highest rated answer (and one which I have sympathy for, broadly speaking) should really be an op-ed blog post somewhere.
As I say in a comment: you can consider categories whose objects arise in analysis and whose morphisms have some flavour of continuity. Hence, category theory can be used in analysis. Fin. Or as Paul Siegel says
What I can say is that many interesting and nontrivial categories do arise in certain parts of functional analysis and it is useful to understand the structure of these categories
So I wanted to start a thread where people could try and convince me otherwise.
I have retitled the question, since it didn't seem that people wanted to answer that specific wording, and since the OP appears not to have meant it literally.
How about posting a lot of similar questions? For example: "Is there a nice application of functional/complex/harmonic analysis to category theory?"
He he, Gerald. I assume you're kidding? Or are you secretly making the point that there really aren't nice applications of category theory to functional analysis?
Although now that I think about it further, it's not an absolutely crazy idea. I mean for example that some result from the theory of metric spaces might suggest a nice generalization to enriched category theory. (If you don't know what I'm talking about, then don't worry about it.)
Here is another of these questions ... http://mathoverflow.net/questions/84306/ ... waiting for a few dozen more ...
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