First, a disclaimer: I am not even close to being an analyst. Second, I don't know of any applications of category theory to the areas of analysis that you mention. I don't think we have got to that point yet, for the reason given below. But here is an answer to a more general question that I hope you'll find illuminating.
I think the thing to remember here is that category theory is 'structural mathematics'. That is, it seeks to understand mathematical objects and constructions purely in terms of abstract external structure, as opposed to internal details about how an object is put together. In areas like algebra and computer science, this sort of structure is already there and visible, so for example it's easy to define the notion of 'group object' or 'monoid action', and to discuss constructions like quotients and semidirect products and so on in purely structural terms.
My impression of analysis is that the structures involved are less unequivocal and less clearly visible, and consequently it's harder to give a broad unified structural picture of the kind that we're used to in algebra. So the category-theoretic or structural understanding of analysis is a good deal less well-developed. But there are some interesting facts about certain structures found in (very elementary) analysis and topology:
- Metric spaces are a particular kind of enriched category, and the notion of Cauchy completeness has a very neat definition in that context.
- Compact Hausdorff spaces are the algebras for a monad on Set (the ultrafilter monad).
- Topological spaces are lax algebras for the same monad on the bicategory Rel of relations.
- Normed vector spaces can be viewed as enriched categories with duals (Lawvere), or as compact-closed ordinary categories equipped with a certain kind of functor (see Geoff Cruttwell's thesis).
- C-star algebras are monadic over Set (although I gather not much more is known about this).
- Non-standard analysis has a nice interpretation in terms of the filter-quotient construction on toposes.
- There is a notion of Fourier or z-transform for Joyal's 'structure types'.
There are probably many more (and this answer is CW so passers-by are invited to add them).
Personally, I think that the structural and material viewpoints on mathematics complement each other very well, so I'd be delighted if someone could point out (or write!) a structural account of -- a sort of 'Mac Lane and Birkhoff' for -- even elementary analysis.