# Hilbert basis of vector space

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.

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## 1 Answer

Consider all orthonormal systems $B$ containing $X$. By definition put $B_1\leq B_2$ if $B_1\subset B_2$. To prove that maximal system (which existence is guaranteed by Zorn's lemma) is a basis consider orthogonal complement of it.

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Thank you very much. – Hiperion Nov 29 '12 at 18:44