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