On a necessary condition for a system of normalized elements to be a basis in a Hilbert space

In this paper we consider a complete, minimal, almost normalized sequence $\{\varphi_k\}^\infty_{k=1}$ of elements of a Hilbert space $H$ such that their inner products have the property $|(\varphi_k,\varphi_j)|\ge\alpha$, $\alpha>0$ for all sufficiently large numbers $k,j$. It was proved that this sequence is not an unconditional basis in $H$.