When is a sequentially closed subspace the kernel of a sequentially continuous linear functional?

Question

Let $E$ be a Banach space with continuous dual $E'$, endowed with the weak$^*$ topology. Let $V\subseteq E'$ be a (weak$^*$-)dense subspace of $E'$. If $V$ is also sequentially closed (with respect to the weak$^*$ topology on $E'$), does there exist a weak$^*$ sequentially continuous linear functional $\phi:E'\longrightarrow\mathbb{R}$, such that $\textsf{ker}(\phi)=V$?