Are there some interesting applications of Banach-Alaoglu theorem in the theory of distributions? The theorem provides compact subsets in the $w^*$-topology, so distributions seem a great place for applications of this theorem.
(I would imagine a typical application to start with a sequence $T_1\supset T_2\supset T_3\supset\dots$ of non-empty compact subsets of $\mathscr D'$, with the compactness of $T_n$ coming from the Banach-Alaoglu theorem; this would imply that $\bigcap_n T_n\neq\emptyset$. However, I don't know how to get an interesting sequence $T_n$, i.e. such that the fact $\bigcap_n T_n\neq\emptyset$ is non-trivial and interesting.)