$H -$ Hilbert space, $D \subset H$ and closure $\overline{span D} = H$. The set $D$ is called dictionary. Let $$\rho(D):= \inf_{x\in H, |x| = 1} \sup_{g \in D} |\langle x,g \rangle|.$$
I need to prove that $\rho(D) = 0 \Leftrightarrow \exists$ orthonormal system $\{ w_n \}$ in $H$ such that $$\lim_{n\rightarrow \infty} \sup_{g\in D} |\langle w_n,g \rangle| = 0.$$
It is obviously from right to left. If the left condition is true then existense of sequence $w_n$ such that the right conditon perfomed is obviously too (by definition of $\inf)$. I need to show that we can find orthonormal system.