If $L$ is a normed space with the property that if $M$ is a hyperplane in $L^*$ and $M \cap \operatorname{ball} L^*$ is weak-star closed $\implies$ $M$ itself is weak star closed, then how do I show $L$ is a Banach space?
I think I should should identify $L$ with $L^{**}$, but I'm not sure how to prove this.