This is a follow up to the question: Biorthogonal functionals. A positive answer to that question implies a negative answer to this one.

If $X$ is a separable Banach space, can we find a basic sequence $(x^{*}_n)$ in $X^{*}$ with the property that for any subsequence $(n_k)$ we have that: $$ \bigcap_{k=1}^{\infty}\ker{x^{*}_{n_k}}=\{0\} $$

Clearly this is not possible in a reflexive space, as the previous question has a positive answer in the reflexive case. Therefore the only case of interest is when $X$ is not reflexive.