Separable quotient problem

The separable quotient problem asks whether every infinite-dimensional Banach space $X$ has a closed subspace $Y$ such that the quotient $X/Y$ is both infinite-dimensional and separable. The question goes back to the early days of Banach space theory around Banach's 1932 monograph [Banach1932OperationsLineaires] and is traditionally associated with Mazur.

For separable $X$ the answer is immediate (take $Y=\{0\}$), so the substance lies in the nonseparable case. There the answer is affirmative for large classes of spaces: reflexive spaces, weakly compactly generated spaces, dual spaces, $C(K)$ spaces and many others, as surveyed in [FerrandoKakolLopezPellicerSaxon2018SeparableQuotient]. Recent work continues to enlarge the known territory, for instance for nonseparable Bourgain–Pisier-type spaces [Patri2026SeparableQuotient].

No construction is known that produces a separable infinite-dimensional quotient of a completely arbitrary nonseparable Banach space, and no counterexample has been found either. The problem therefore remains open in full generality: a resolution requires either a genuinely general construction or an exotic space all of whose infinite-dimensional quotients are nonseparable.