# Weak* Topology coarser than topology of uniform convergence on compact sets

Let $$E$$ be a locally convex topological space. Is the weak$$^*$$-topology on its topological dual space $$\sigma(E', E)$$ coarser than the topology of uniform convergence on compact sets?

I know that the topology of uniform convergence on compact sets $$\tau_c$$ is induced by the family $$(p_S)_{S \in \mathcal S}$$ of seminorms $$p_S:f \mapsto \sup_{x \in S} f(x)$$ where $$\mathcal S$$ denotes the set of all compact subsets of $$E$$. Can I choose a similar set for the weak$$^*$$-topology? Maybe all singleton sets?

• The weak^* topology is the topology of point-wise convergence. Commented Jun 16, 2021 at 5:53
• You should use uniform convergence on bounded sets and not compact sets. For many $E$'s of interest though you will not see the difference because they are Montel. Commented Jun 25, 2021 at 20:10

Yes. $$\sigma(E^*,E)$$ is determined by the evaluation-map seminorms $$\{(f\mapsto|f(x)|):x\in E\}$$ This is a strict subset of the seminorms for the compact-open topology, since each $$\{x\}\in\mathcal{S}$$ and $$\sup_{y\in\{x\}}{|f(y)|}=|f(x)|$$