A set modelling the real numbers is defined in introductory math courses. The process of quotienting out all the set-theoretic slag to arrive at the real numbers, singular, is generally skipped. As well it should be: it's too ugly to be an end in itself, and too subtle to be of use to freshmen.

> Hilbert space? A complete inner product space where complete means every Cauchy convergent sequence converges.

I said rigged Hilbert spaces. The proper setting for QM is a Hilbert space equipped with dense subspace P of test functions (for which the inclusion P -> H is continuous). This induces an inclusion of dual spaces H* -> P*, and Reisz-Frechet gives us a natural isomorphism H ~ H*, so we have the "Gelfand triple": P -> H -> P*.

The view from 10,000 feet here is that P describes the space of possible measurements, P* describes the distributions, and the fact that the inner product <P, P*> factors through H is a compactness condition. So in particular, a plane wave |k> lives in P*, not H.

But you can't teach this to introductory physics students. They won't have the necessary math background for several years. (Many of them will never have it). And yet they have to be taught.