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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?

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  • $\begingroup$ The weak^* topology is the topology of point-wise convergence. $\endgroup$ Commented Jun 16, 2021 at 5:53
  • $\begingroup$ 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. $\endgroup$ Commented Jun 25, 2021 at 20:10

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

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  • $\begingroup$ Thanks a lot!!! $\endgroup$
    – Antigone
    Commented Jun 16, 2021 at 6:02

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