Dense Subspace: ONB

Question

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...)

Answer 1

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).