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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 $E \subset X$ 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 every such $E$ nonmeager?

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?

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 every such $E$ nonmeager?

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?

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 every such $E$ nonmeager?

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?

added 168 characters in body
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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 every such $E$ nonmeager?

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?

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 every such $E$ nonmeager?

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

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 every such $E$ nonmeager?

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?

unnecessary words
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Nate Eldredge
  • 27.2k
  • 15
  • 20

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.

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.

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 every such $E$ nonmeager?

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

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Nate Eldredge
  • 27.2k
  • 15
  • 20
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