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 $H$ be a Hilbert space and $(e_n)_{n=1,2,\ldots}$ be a complete orthonormal sequence in $H$. We want to show that if $a_{np}=(e_n,f_p)$ then $\sum_{p=1}^{\infty}a_{np} \overline{a_{mp}}=\delta_{nm}$ and $\sum_{n=1}^{\infty}a_{np}\overline{a_{nq}}=\delta_{pq}$ where $\delta_{ij}$ is the Kronecker delta and $(f_p)_{p=1,2,\dots}$ is an other complete orthonormal sequence in $H$..

I've been thinking about this for a while now - any thoughts / hints about where to start? Thanks!

share|improve this question

1 Answer 1

I guess $(f_p)_{p\ge1}$ must be another complete orthonormal sequence.

In that case, you ought to be able to show that $\sum_{p=1}^{\infty}a_{np} \overline{a_{mp}}=(e_n,e_m)$ and $\sum_{n=1}^{\infty}a_{np}\overline{a_{nq}}=(f_q,f_p)$. By orthonormality, these inner products are $\delta_{nm}$ and $\delta_{pq}$.

Hint: you can use the fact that $x=\sum_{n=1}^\infty (x,e_n)e_n=\sum_{p=1}^\infty (x,f_p)f_p$ for any vector $x\in H$.

share|improve this answer
    
Thank you very much! –  Carlotto May 13 '11 at 17:51

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.