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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!

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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 $x\in H$.

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