Dense Subspace: ONB This might be a duplicate. If so, then please let me know. Thanks!
Given a Hilbert space $\mathcal{H}$.
Consider a dense subspace $\overline{Z}=\mathcal{H}$.
Then it provides an ONB: $\mathcal{S}\subseteq Z$
(I guess it can be shown by slightly adjusting the usual proof via Zorn's lemma...)

I just realized that this cannot be true for the following reason:
Assume every dense subspace would provide an ONB for the Hilbert space:
$$\overline{\mathcal{S}}=\mathcal{H}\quad(\mathcal{S}\subseteq Z)$$
Then, it would serve as well as an ONB for the subspace:
$$\overline{\mathcal{S}}\supseteq Z$$
But there are preHilbert spaces which do not admit any ONB (see Bourbaki or Robert Isreal).
