Weak *-topology of $X^*$ is metrizable if and only if ...

Let $X$ be a topological vector space on which $X^*$ separates points. Prove that "the weak *-topology of $X^*$ is metrizable if and only if $X$ has a finite or countable Hamel basis"?

(A set $\beta$ is a Hamel basis for a vector space $X$ if $\beta$ is a maximal linearly independent subset of $X$. In other words $\beta$ is a Hamel basis if every $x\in X$ has a unique representation as a finite linear combination of element of $\beta$.)