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Nate Eldredge
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Self-promotion, but I would like to mention my question Meager 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 $$E \subset X$$ determines weak- convergence* (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 it true that every such $$E$$ is nonmeager?

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

Nate Eldredge
• 27.5k
• 16
• 20