Let $X$,$Y$ be locally convex topological vector spaces. Assume now I have an operator $T:Y'\rightarrow X'$ where $Y'$ and $X'$ are equipped with the weak$^*$-topology.
Does this imply that $T$ is continuous, iff for every sequence $f_n,f \subset Y'$ such that $f_n(y) \rightarrow f(y)$ for every $y \in Y$ we have $ T(f_n)(x) \rightarrow T(f)(x)$ for every $x \in X$?
This seems natural, but I don't know why this sequential continuity is sufficient for continuity in the weak$^*$-topology?