Consider the following result:
Let $E$ be a Banach space, and let $X\subseteq E^*$ be a subspace. For any $\mu$ in the weak$^*$-closure of $X$ we can find a net $(\mu_\alpha)$ in $X$ which converges to $\mu$, uniformly on compact subsets of $E$.
I think this would follow from considering the Arens-Mackey theorem; but I don't really understand Mackey topologies, if I'm honest. I can write down a proof which uses the Krein-Smulian theorem (the one which says that a convex subset of $E^*$ is weak$^*$-closed if and only if intersections with closed balls are weak$^*$-closed). (Indeed, let $Y$ be the collection of $\mu\in E^*$ which are limits of functionals in $X$ with convergence uniformly on compacta in $E$. Then $Y$ is a subspace, and it is enough to show that $Y$ is weak$^*$-closed. Using the Krein-Smulian theorem we reduce to showing that the set $\{\mu\in Y : \|\mu\|\leq 1\}$ is weak$^*$-closed, which is easy, as we have now reduced the problem to bounded nets in $E^*$).
Is there an easier way to show this result? Is this result actually well-known (e.g. I could just quote a book?)