[Edit:] Partly in response to @Yemon Choi's comments... perhaps nowadays "functional analysts" no longer neglect practical categorical notions, but certainly Rudin and Dunford-Schwartz's "classics" did so. I realize in hindsight that this might have been some "anti-Bourbachiste" reaction. Peter Lax's otherwise very useful relatively recent book does not use any categorical notions. Certainly Riesz-Nagy did not. Eli Stein and co-authors's various books on harmonic analysis didn't speak in any such terms. All this despite L. Schwartz and Grothendieck's publications using such language in the early 1950s. Yosida? Hormander?
I do have a copy of Helemskii's book, and it is striking, by comparison, in its use of categorical notions. Perhaps a little too formally-categorical for my taste, but this isn't a book review. :)
I've tried to incorporate a characterize-rather-than-construct attitude in my functional analysis notes, and modular forms notes, Lie theory notes, and in my algebra notes, too. Oddly, though, even in the latter case (with "category theory" somehow traditionally pigeon-holed as "algebra") describing an "indeterminate" $x$ in a polynomial ring $k[x]$ as being just a part of the description of a "free algebra in one generator" is typically viewed (by students) as a needless extravagance. This despite my attempt to debunk fuzzier notions of "indeterminate" or "variable". The purported partitioning-up of mathematics into "algebra" and "analysis" and "geometry" and "foundations" seems to have an unfortunate appeal to beginners, perhaps as balm to feelings of inadequacy, by offering an excuse for ignorance or limitations?
To be fair (!?!), we might suppose that some tastes genuinely prefer what "we" would perceive as clunky, irrelevant-detail-laden descriptions, and, reciprocally, might describe "our" viewpoint as having lost contact with concrete details (even though I'd disagree).
Maybe it's not all completely rational. :)