Lemma 41.10.1.label Let $E, F$ be locally convex spaces of $K \in \RC$, $B \subset L(E; F^{*})$ be equicontinuous, then the following two topologies on $B$ coincide:
- (1)
The weak* topology under the identification $L(E; F^{*}) = (E \wh \otimes_{\pi} F)^{*}$.
- (2)
The topology of pointwise convergence on $\bracsn{e \otimes f|e \in E, f \in F}$.
Proof. As $B$ is equicontinuous, the Arzelà-Ascoli Theorem implies that the weak* topology coincides with the topology of pointwise convergence on any dense subset. By linearity, the topology of pointwise convergence on $\bracsn{e \otimes f|e \in E, f \in F}$ coincides with the topology of pointwise convergence on $E \otimes F$. As $E \otimes F$ is dense $E \wh \otimes_{\pi} F$, the topology of pointwise convergence on $E \otimes F$ coincides with the topology of pointwise convergence on $\bracsn{e \otimes f|e \in E, f \in F}$.$\square$
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