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 $\beta$.)

Solution 1:

This should be obvious from the characterization of the weak-$^*$ topology as the topology of pointwise convergence and it's easy to show that $\mathbb R^A$ is only metrizable when the cardinality of $A$ is countable. But by the Hahn-Banach theorem $X^*$ has a Hamel basis that's at least the same size as that of $X$ (assuming that $X$ is locally convex. $L^p$ for $0<p<1$ has no non-zero continuous linear functionals) so $X^*$ cannot be metrizable.

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