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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.

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    Yes. Choose an orthonormal basis of the orthogonal complement of $V$ in $U$ and add it to the given orthonormal basis of $V$. – 2017-01-16
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    Ok @Jochen thanks so much – 2017-01-16

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