Let $E$ be a reflexive Banach space. Let $\{x_n\}_n$ be a bounded sequence of linearly independent elements of $E$. Does there exist a sequence $\{\phi_n\}_n$ of elements of $E^*$ (the dual of $E$) such that $\langle x_n,\phi_m\rangle=\delta_{mn}$ and $\sum\|\phi_n\|<\infty$.