Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question

1 Answer 1

up vote 1 down vote accepted

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.

share|improve this answer
    
Thank you very much. –  Hiperion Nov 29 '12 at 18:44

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.