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. (Lean)

Manuscript

Relative independence of the separable quotient problem

September 23, 2026 · source · cite as \cite{OAI:Relative-independence-of-the-separable-quotient-problem-September-23-2026}

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.

Result 323 · Functional analysis. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.