Let $T$ be a linear transformation of a Banach space $B$ into a Banach Space $B'$. Let $E \subset B^{'*} $ satisfying -
A) $E$ separates points of $B'$ (i.e. for for $x , y \in B' , x \neq y$ there is some $\phi \in E$ such that $\phi (x) \neq \phi(y)$) .
B) For every $\phi \in E$, $\phi \circ T$ is continuous.
I need to show that $T$ is continuous.
Now I suppose I need to use the Closed Graph theorem here but I am not getting any clue as to where to go with it.