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The separable quotient problem for Banach spaces

Theorem 1.1: for $\mathbb F\in\{\mathbb R,\mathbb C\}$ , ZFC proves that if $2^{\aleph_0}$ is real-valued measurable then every infinite-dimensional Banach space over $\mathbb F$ has a separable infinite-dimensional quotient, and that under CH there is a counterexample of density $\aleph_1$ . So the problem is independent of ZFC if ZFC plus a measurable cardinal is consistent; restricted to density $\aleph_1$ it is independent relative to ZFC alone (Corollary 1.2, with Martin's axiom). It does not give a ZFC answer, does not remove the measurable-cardinal strength for the unrestricted statement, and leaves open which other axioms decide it.

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Occurred: Sep 23, 2026

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Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The separable quotient problem for Banach spaces · Separable quotient problem

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VibeMathed record: The separable quotient problem for Banach spaces

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the problem. ZFC proves (i) if the continuum is real-valued measurable, every infinite-dimensional Banach space over $\mathbb R$ or $\mathbb C$ has a separable infinite-dimensional quotient, and (ii) under CH some space has none; hence independence relative to a measurable cardinal, and full independence for density $\aleph_1$ relative to ZFC. The challenge lean/ComparatorChallenges/SeparableQuotientNegative.json and its solution module OAI.Analysis.SeparableQuotients.Main exist at the pinned commit (found via lean/docs/323.md) but are not in the formalization catalogue. The statement was read here: CH implies the separable-quotient assertion fails over both $\mathbb R$ and $\mathbb C$ . That is narrower than the headline: the positive direction and the independence conclusion are not formalised. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice. The independence result is relative to the consistency of a measurable cardinal. Listed as Unreviewed rather than Lean-checked because its formal statement covers only one direction of the independence claim (CH implies a counterexample).