Let $(V, \lVert \cdot \rVert)$ be a separable Banach space and let $B_{V^*}$ denote the closed ball in the dual $V^*$. Suppose we have a family $C \subseteq B_{V^*}$ such that $\bigcap_{\Lambda \in C} \mathrm{ker}(\Lambda) = \{ 0 \}$. Is it always possible to find a subfamily $\widetilde{C} \subseteq C$ such that $\bigcap_{\Lambda \in \widetilde{C}} \mathrm{ker}(\Lambda) = \{ 0 \}$, but removing any element of $\widetilde{C}$ makes this equality false?