Let $E, F$ be locally convex $\mathbb{K}$ vector spaces and $T: E\to F$ linear and continuous. Let $T': F'\to E'$ be defined through $T'(g)=g\circ T$
$T': F'\to E'$ is weak-$\ast$-continuous (hence continuous when $F'$ and $E'$ are provided with the weak-$\ast$ topology)
This question is related to $T: E\to F$ weak continuous, $E, F$ locally convex and unfortunatly I do not know where to start with the proof. I struggle with the definition of the weak topology and the weak-$\ast$-topology.
Can you help me, or give me a hint. Thanks in advance.
Assume $g_{\delta}\rightarrow g$ in $w^{\ast}$ of $F'$, and let $x\in E$, then $T(x)\in F$ and hence $|T'(g_{\delta})(x)-T'(g)(x)|=|g_{\delta}\circ T(x)-g\circ T(x)|=|g_{\delta}(T(x))-g(T(x))|\rightarrow 0$.