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?
Best Answer
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 $$\sup_{y\in\{x\}}{|f(y)|}=|f(x)|$$