Let $V$ a vector space with inner product and $X\subset V$ orthonormal. Prove that exists a Hilbert basis (an orthonormal set of vectors with the property that every vector in $V$ can be written as an infinite linear combination of the vectors in the basis) such that $X\subset B$.
I can consider $B=X\cup X^{\perp}$ and this set will be maximal, but I am not sure about this, is it correct?
Another idea is using the Zorn lemma, but I need a "chain", how can I build this chain?
Thanks for your help.