Dimensionality of a Hilbert Space: Countable or Uncountable? [duplicate]

(My question is different than this one and the similar one about the free particle, so hold back on casting a close vote, please).

So, I was reading on Wikipedia, and ran into this statement in the Dirac-von Neumann axioms: The space $$\mathbb{H}$$ is a fixed complex Hilbert space of countable infinite dimension.

This is rather confusing to me; I use position and momentum as bases for this Hilbert space all the time, and those bases have cardinality $$2^{\aleph_0}$$. How is it that the space is countably infinite when we use uncountable basis sets to perform basic calculations?

• – SRS Jun 14 '20 at 2:43

$$L^2(\mathbb R)$$ is indeed separable, however - the weighted Hermite polynomials (which are incidentally the eigenstates of the quantum harmonic oscillator) constitute a countable basis for it.