Separability of a Hilbert space and its implications for the formalism of QM In the text I'm using for QM, one of the properties listed for Hilbert space that is a mystery to me is the property that it is separable. Quoted from text (N. Zettili: Quantum Mechanics: Concepts and Applications, p. 81):

There exists a Cauchy Sequence $\psi_{n} \ \epsilon \ H (n = 1, 2, ...)$ such that for every $\psi$ of $H$ and $\varepsilon > 0$, there exists at least one $\psi_{n}$ of the sequence for which
  $$ || \psi - \psi _{n} || < \varepsilon.$$

I'm having a very hard time deciphering what this exactly means. From my initial research, this is basically demonstrating that Hilbert space admits countable orthonormal bases. 


*

*How does this fact follow from the above? 

*And what exactly is the importance of having a countable orthonormal basis to the formalism of QM? 

*What would be the implications if Hilbert space did not admit a countable orthonormal basis, for example?
 A: I usually see it in the reverse way, but it is a matter of taste. Hilbert spaces, in general, can have  bases of arbitrarily high cardinality. The specific one used on QM is, by construction, isomorphic to the space L2, the space of square-integrable functions. From there you can show that this particular Hilbert space is separable, because it is a theorem that a Hilbert space is separable if and only if it has a countable orthonormal basis, and L2 has one. 
A: As showed by Solovay here, in a non-separable Hilbert space $H$ there may be probability measures that cannot be written, for any $M$ closed subspace of $H$, as $\mu (M)=\mathrm{Tr}[\rho \mathbb{1}_M]$, for some positive self-adjoint trace class $\rho$ with trace 1 (density matrix). Here $\mathbb{1}_M$ denotes the orthogonal projection on $M$. [The proof of the existence of such "exotic measures" is undecidable in ZFC, however it is equivalent to the existence of a (real valued) measurable cardinal]
In some sense it means that in non-separable Hilbert spaces there may exist analogues to "normal quantum states" that are not density matrices$^\dagger$.
Remark: For normal quantum state I mean a ultraweakly positive continuous functional on the $C^*$-algebra of bounded operators that is interpreted to give their expectation value.
$^\dagger$: I mean that even if these states are probability measures with the countable additivity of orthogonal closed subspaces property, are not expressed as the trace of density matrices (while on separable Hilbert spaces this is always the case, and these measures are in one to one correspondence with normal states).
