Orthonormal Bases in a Hilbert Space and Vector Subspaces Let $H$ be a Hilbert space and $S$ a vector subspace of $H$ which is dense in H. Does there exist an orthonormal basis $(u_{\alpha})_{\alpha \in A}$ of $H$ such that $\{ u_{\alpha} :    \alpha \in A \} \subseteq S$?
I think the answer is generally negative, but I can't find a counterexample. Any help is welcome.

There is a post here that can be useful. 
