Independence of the separable quotient problem. Establishes, relative to the consistency of a measurable cardinal, that the separable quotient problem is independent of ZFC. The assertion that every infinite-dimensional Banach space has a separable infinite-dimensional quotient can hold for all real and complex Banach spaces, whereas the continuum hypothesis yields counterexamples over both fields.
released 2026-09-23 | 5 theorems · 19 lemmas · 31 proofs · 17,894 words |
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The separable quotient problem asks whether every infinite-dimensional Banach space has a separable infinite-dimensional quotient. We prove that this statement is independent of ZFC, relative to the consistency of ZFC with a measurable cardinal. This holds over both the real and complex fields. For spaces of norm density ℵ1, we obtain independence relative to the consistency of ZFC alone.