Let $X$ be a Banach space and let $E$ be a subspace of the dual $X^*$. Suppose that $y$ belongs to the weak* closure of $E$.
Does there exists a (norm) bounded net $(y_i)$ which converges to $y$ for the weak* topology ?
I think that it is related to the Krein-Smulian theorem but it is not clear for me.
If we can choose a sequence instead of a net, it is immediate (a bounded weak* convergent sequence is bounded) but I does not believe that it is possible.