I want to prove that, in a RKHS (Reproducing Kernel Hilbert Space), being k(t,s) its reproducing kernel:
Orthonormal system $\{ e_n(t)\}_{n=1}^{\infty}$ is complete $\Leftrightarrow$ $k(t,t) = \sum_{n=0}^{\infty}{|e_n(t)|^2}, \forall t \in \Omega$.
So far, I've proved that if $\{ e_n(t)\}_{n=1}^{\infty}$ is complete, then it is an orthonormal basis, and the Parseval identity applies, giving:
$$||k_t||^2 = k(t,t) = \sum_{n=1}^{\infty}{|\langle k_t,e_n\rangle|^2}= \sum_{n=1}^{\infty}{|\langle e_n,k_t\rangle|^2} = \sum_{n=1}^{\infty}{|e_n(t)|^2}.$$
The last identity is given by the reproducing property.
My problem is in the converse proof: given $k(t,t) = \sum_{n=0}^{\infty}{|e_n(t)|^2}, \forall t \in \Omega \Rightarrow \{ e_n(t)\}_{n=1}^{\infty}$ is complete.