Suppose $X$ is a normed space and that the only hyperplane $M$ in $X^*$ such that $M∩ball(X^*)$ is weak star closed are those that are weak-star closed. Show $X$ is Banach space.
(Here, $(^*)$ = closed unit ball in $X^*$)
My attempt: Identify $X$ with a subset of $X^{**}$. Let $x_n$ be Cauchy in $$. Since $^{**}$ is Banach space, it converges to some $x^{**}\in X^{**}$. If $ker(x^{**}) \cap ball(X^{*})$ is weak-star closed, then $ker(x^{**})$ is weak-star closed by the hypothesis. But then this implies $x^{**} \in (X^*, wk^*)^* = X$, by this . Thus we're done.
Hence, it suffices to show $ker(x^{**}) \cap ball(X^{**})$ is weak-star closed, but I have no idea how to prove this. Thanks for your helps!
