Let $X$ be a Banach space, $X^*$ its dual. Suppose $E$ is a linear subspace of $X^*$ which separates points (i.e. if $f(x)=0$ for all $f \in E$, then $x=0$).
Must $E$ be weak-* dense in $X^*$?
In all the examples I can think of, it is, but this seems too good to be true.
If not, does it help if $X$ is separable? Reflexive?