So there's a couple things going on.

First, you glazed right over the 'rigged' in hither_shore's original post. Did you just not see it or did you think it was some kind of meaningless term to be ignored? Either way, in the context of 'really stupid mistakes' it stands out.

Second, it's not very productive to be so uncharitable when reading technical work. As a reader, you can either assume 'the author is an idiot, I'm going to stop here' or 'the author has elided some details, and they will probably not be relevant later, so I will intuit what they should have said and carry on'.

In this particular case, I think it's clear that you should have interpreted 'Hilbert space of differentiable functions' as something like 'the completion of the pre-Hilbert space of differentiable functions' aka 'the smallest Hilbert space containing the differentiable functions'.

That's the first case of rigged I ever saw.

For your "pre-Hilbert" space, I never saw that either but likely my guess at what you mean would be correct.

Can go ahead and take the completion (maybe the same as what used to be called the "normal completion"), but the result will have a lot of functions that are not differentiable and, thus, not quantum mechanics wave functions. So, I would be reluctant to take the resulting actual Hilbert space as what physics was talking about.

If we are going to talk seriously about a subject as serious as quantum mechanics, e.g., to try to understand what is known and maybe to try to clean up what is commonly called its many loose ends and maybe make progress in some applications, e.g., condensed matter physics, super conductivity, maybe even something as common as just why a grid of wire reflects radar waves, maybe connecting with gravity, dark energy, the first second, the Casimir effect, Hawking radiation, and I would include entanglement, then IMHO we need to be as clear as possible on the definitions, theorems, proofs, passing between physics and pure math, etc. After all, often in the research, we are trying to replace some of what is accepted or find what in the past was not discovered. When I see sloppy, I'm not making progress but looking at some at least some little research problems.

> but the result will have a lot of functions that are not differentiable and, thus, not quantum mechanics wave functions. So, I would be reluctant to take the resulting actual Hilbert space as what physics was talking about.

Wavefunctions aren't differentiable functions, because they're not functions. They're distributions - described in general by the rigged Hilbert space structure I described previously, and in the simplified normalizable case typical of introductory courses by L^2(R^n).

> to try to understand what is known and maybe to try to clean up what is commonly called its many loose ends and maybe make progress in some applications, e.g., condensed matter physics, super conductivity, maybe even something as common as just why a grid of wire reflects radar waves, maybe connecting with gravity, dark energy, the first second, the Casimir effect, Hawking radiation, and I would include entanglement, then IMHO we need to be as clear as possible on the definitions, theorems, proofs, passing between physics and pure math, etc.

Ok, but that's not what undergraduate textbooks are for. If you want mathematical physics without any pedagogical concessions, go read mathematical physics papers.