Let $T\colon X\to Y$ be a continuous linear surjection between Frechet spaces. Then it is possible to use Michael's selection theorem to show that there exists a continuous (non-linear) function $g\colon Y\to X$ such that $g\circ T = {\rm id}_X$.

I am wondering it is possible to obtain a weak-continuous counterpart of this theorem. Namely, what if $X$ and $Y$ are dual Frechet spaces and $T$ is weak-continuous. Can we produce such a $g$ that would be weak*-continuous?

The relevant result I quoted is Corollary 7.1 in Bessaga and Pełczyński's, Selected topics in infinite-dimensional topology.