Let $H$ be a separable, infinite dimensional, complex Hilbert space
Let $S \subset T \subset H$ be two subsets of $H$ and the $H$ subspaces $V=\overline{span(S)}$ and $U=\overline{span(T)}$
I would like to know if is it possible to find the orthonormal basis $SO$ of $V$ and the orthonormal basis $TO$ of $U$ such that $SO \subset TO$
Thanks for any suggestion.