weak convergence star $wk^*$

Question

I'm starting to study the weak star ( $wk^*$) topology and I want to solve the following task:

Let $X$ banach space and $F\in X^{**}$ (bidual space) such that $Ker(F)$ is $wk^*$ closed then $F$ is $wk^*$ continuous.

We are working with the book of conway, which states that the above is true for Banach-Alaouglu theorem. Unfortunately I cannot find how to apply this theorem. I would appreciate any suggestions.

Answer 1

Maybe I miss something obvious, but in every topological vector space, a linear functional is continuous if and only if its kernel is closed. See for example: The kernel of a continuous linear operator is a closed subspace? Hence, $F$ is continuous in the wk*-topology if and only if its kernel is wk*-closed.