In relatively mundane, but intensely useful and practical, ways, the naive-category-theory attitude to characterize things by their interactions with other things, rather than to construct (without letting on what the goal is until after a sequence of mysterious lemmas), is enormously useful to me.
E.g., it was a revelation, by now many years ago, to see that the topology on the space of test functions was a colimit (of Frechet spaces). Of course, L. Schwartz already worked in those terms, but, even nowadays, few "introductory functional analysis" books mention such a thing. I was baffled for some time by Rudin's "definition" of the topology on test functions, until it gradually dawned on me that he was constructing a thing which he would gradually prove was the colimit, but, sadly, without every quite admitting this. It is easy to imagine that it was his, and many others', opinion that "categorical notions" were the special purview of algebraic topologists or algebraic geometers, rather than being broadly helpful.
Similarly, in situations where a topological vector space is, in truth, a colimit of finite-dimensional ones, it is distressingly-often said that this colimit "has no topology", or "has the discrete topology", ... and thus that we'll ignore the topology. What is true is that it has a unique topology (since finite-dimensional vector spaces over complete non-discrete division rings such as $\mathbb R$ or $\mathbb C$ do, and the colimit is unique, at least if we stay in a category of locally convex tvs's). Also, every linear functional on it is continuous (!). But it certainly is not discrete, because then scalar multiplication wouldn't be continuous, for one thing. But, despite the prevalence of needlessly inaccurate comments on the topology, the fact that all linear maps from it to any other tvs are continuous mostly lets people "get by" regardless.
Spaces with topologies given by collections of semi-norms are (projective/filtered) limits of Banach spaces. Doctrinaire functional analysts seem not to say this, but it very nicely organizes several aspects of that situation. An important tangible example is smooth functions on an interval $[a,b]$, which is the limit of the Banach spaces $C^k[a,b]$. Sobolev imbedding shows that the (positively-indexed) $L^2$ Sobolev spaces $H^s[a,b]$ are {\it cofinal} with the $C^k$'s, so have the same limit: $H^\infty[a,b]\approx C^\infty[a,b]$, and such.
All very mundane, but clarifying.
[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. :)
Best Answer
John Stillwell's recent book Naive Lie Theory is amazing and in a similar vein. It provides great geometrical intuition for many of the common matrix groups. What is particularly impressive about this book is how he motivates more complicated ideas, such as maximal tori, in a very elementary fashion. It is perfect for undergrads looking for a good introduction.