When I read the prove of the Goldstine Theorem(See An Introduction to Banach Space Theory Robert E. Megginson 2.6.26), I find it use the separation theorem for the w* topology without any details.
Suppose $X$ is a normed space, $F$ is a w* closed convex set of $X^*$, $x^*$ is in $X^*$ but not in $F$, then there is $x \in X$, such that $|x^* (x)|> \sup \{\Re y^*(x) \mid y^*\in F\}$.
Can you prove it?