One of the very important things to know is that, while an infinite-dimensional, separable Hilbert space is basically "n-dimensional space, but more so", there

One of the very important things to know is that, while an infinite-dimensional, separable Hilbert space is basically "n-dimensional space, but more so", there are more general kinds of infinite-dimensional spaces (I'm thinking of Banach spaces, but you can certainly get still more general than that) that are much more general than that. The prototypical examples of these are the L^p spaces, where p stands for ∞, or a real number p ≥ 1—but p isn't the dimension, as in ℝ^n (they're all infinite dimensional, at least for a reasonable underlying metric space). Rather, it's a parameter that controls, in some sense, how close the geometry is to being governed by the Pythagorean theorem, so that only p = 2 gives (a Hilbert space, and hence) the 'usual' geometry. I think your point with functional programming is spot on. Just as one learns, to pick that bête noire, monads not by reading yet another clever re-packaging that somehow only manages to make them sounds more difficult, but rather by finding a problem for which they're relevant and getting a feel for them by using them, so too does one learn about infinite-dimensional space not by treating it as some sort of philosophical profundity, but by finding a problem for which it's the right setting, and realising that it's just a mathematical tool like any other.