Let $E, F$ be locally convex $\mathbb{K}$ vector spaces and $T: E\to F$ linear and continuous. Show, that $T: E\to F$ is "weak continuous", hence continuous if $E$ and $F$ are provided with the weak topology.
[Note that this task is not complete. There are more tasks to it, so there might be assumtions, which are not needed]
I am struggeling generally with the weak topology. I am not sure how I can best show, that $T$ is continuous regarding the weak topology. I would try to show, that the preimage of an open set, is open.
Is it correct, that I could also show, that $T$ is continuous in $0$, since $E, F$ are locally convex $\mathbb{K}$ vector spaces (espacially topological $\mathbb{K}$ vector spaces) and $T$ is linear, this should be equivalent. Am I right?
Can you help me, because I do not know how to work with the weak topology here. Thanks in advance.
This is similar to your other post: For $x_{\delta}\rightarrow 0$ in $w$ of $E$, we need to show $T(x_{\delta})\rightarrow 0$ in $w$ of $F$. So fix a $g\in F'$, then $g\circ T\in E'$ and hence $|g\circ T(x_{\delta})|\rightarrow 0$, but $|g\circ T(x_{\delta})|=|g(T(x_{\delta}))|$.