Self-promotion, but I would like to mention my question Meager subspaces of a Banach space and weak-* convergenceMeager subspaces of a Banach space and weak-* convergence. It contains two questions. I have resolved Q1 and posted it as an answer (hence the question is no longer "unanswered") but Q2 has not been resolved. I quote it here for visibility.

Q2.Let $X$ be a Banach space. Let us say a linear subspace $E \subset X$* (of sequences) if for every sequence $\{f_n\} \subset X^*$ such that $f_n(x) \to 0$ for every $x \in E$, we have $f_n(x) \to 0$ for every $x \in X$. Is every such $E$ nonmeager?determines weak-convergence

The converse is an easy exercise with the uniform boundedness principle.

**Update** Q2 has now been resolved. The answer is No (though I would still be interested in a separable counterexample). Should I delete this?