 Solution

The set $K$ is weak-star compact by the Banach-Alaoglu theorem and weak-star metrizable by part a. Choose a decreasing countable neighbourhood base $(U_n)$ at $f$. Since $f\in K_{\epsilon }^{\prime }$, the set $U_n\cap K$ cannot lie inside the closed norm ball of radius $\epsilon /2$ about $f$, for then its diameter would be at most $\epsilon$. ChooseThe neighbourhood-base property gives $g_n\stackrel{w^{\ast }}{\to }f$.

Solved by gpt-5.6-sol high.