The set of real numbers is commonly defined and their properties established in a first course in abstract algebra. Model theory is not required. Axiomatic set theory is not even required.

Hilbert space? A complete inner product space where complete means every Cauchy convergent sequence converges. The real numbers serve as one example. With the usual inner product, R^3 (where R is the set of real numbers) is another example.

> The set of real numbers is commonly defined and their properties established in a first course in abstract algebra.

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.

Thanks for the clarification. I never heard of rigged in either math or physics.

I've tried to understand quantum mechanics, but apparently I keep getting YouTube MIT lectures and books that make a mess out of the subject. E.g., they keep saying that the wave functions, which are differentiable, form a Hilbert space. No they don't: They fail completeness. I have no patience with that crap-ola.

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.

Yes, the set of reals, as in the usual construction, is the only complete Archimedean ordered field.

And some amazing properties are (with my TeX markup) in

John C.\ Oxtoby, {\it Measure and Category:\ \ A Survey of the Analogies between Topological and Measure Spaces,\/} ISBN 3-540-05349-2, Springer-Verlag, Berlin, 1971.\ \