If a subspace of $X^*$ is weak*-dense, does it separate points?

Question

Here $X$ is some normed space.

I know the converse is true, but I don't know a proof for the other direction.

That is, if $F\subset X^*$ is a subspace that is weak*-dense how would one show that $F$ separates points (i.e. if $f(x)=0$ for all $f\in F$ then $x=0$)?

Answer 1

Suppose that $f(x)=0$ for all $f\in F$.

Let $g\in X^*$. Then there exists a net $\{f_j\}\subset F$ with $f_j\to g$ weak$^*$. So $$ g(x)=\lim f_j(x)=\lim 0=0. $$ Thus $g(x)=0$ for all $g\in X^*$, and then $x=0$.