How is the weak-star topology useful?

Question

Today I learnt something about the weak-star topology, but I don't know what the use of weak-star topology is. I hope someone can tell me what we can do with the weak-star topology. Thanks in advance!

Answer 1

The main use of weak* topology is to provide a topolgy on $V^*$ for a normed space(or a TVS) $V$ such that the unit ball in $V^*$ is compact, which is the Banach-Alaoglu Theorem. There are numerous places in analysis where we use this topology, for example from Riesz Representation theorem, for any locally compact Hausdorff space $X$, if $\mathcal{M}(X) $ is the space of complex Radon measures, then the dual $ C_0(X)^* \cong \mathcal{M}(X) $, So from usual sup norm topology of $ C_0(X) $ you can induce the weak* topology on Radon measures, thus you can have convergence, compactness results in measures. If you are familiar with distribution theory, you find that given the inductive limit topology on $ C^\infty_c(\Omega) $, weak* topology is induced on the space of distributions. This is a fundamental requirement in PDEs.