Theorem: Let $X$ be a normed linear space. Then weak topology on $X$ is metrizable if and only if $X$ is finite dimensional.
I have proven the forward direction, which is, if weak topology on $X$ is metrizable, then $X$ is finite dimensional.
To prove another direction, I think it suffices to prove that if $X$ is finite dimensional, then weak and norm topologies coincide, as norm topology is induced by a metric.
However, I would like to have a proof of defining a metric $d$ on weak topology explicitly.