On some systems of a Hilbert space which are not bases

In this paper we consider a sequence of normalized vectors $\{h_n\}^\infty_{n=1}$ in a Hilbert space $H$ such that the inner products $\langle h_i,h_j\rangle\ge\alpha$, $\alpha>0$, $i\ne j$, $i,j\in\mathbf N$. It is shown that this sequence of vectors is not a base in $H$.