Mazur’s separable quotient problem

Think of a space with infinitely many independent directions. A quotient forgets a closed collection of directions by treating vectors that differ along them as the same. Can this always leave infinitely many directions, yet a space approximable from a countable list?[1] [2]

For square-summable coordinates indexed by an uncountable set, keeping countably many coordinates gives an infinite-dimensional separable quotient. Can every infinite-dimensional complete normed space admit such a continuous linear image when it has no convenient coordinates? Selecting a separable subspace alone does not create an onto map from the whole space.

For every infinite-dimensional Banach X over ℝ or ℂ, seek closed Y⊂X with dim(X/Y)=∞ and X/Y norm-separable. A bounded onto map T:X→Z supplies Y=kerT; the open mapping theorem identifies the quotient with Z. Separability here is norm separability, not merely a weaker-topology property.[1] [2]

The answer is easy when the original space already has a countable dense list: forget nothing. The difficult case is an arbitrary nonseparable Banach space. Finding a smaller separable space inside it does not provide the required quotient.[1] [2]

The general nonseparable problem remains outside established dual/reflexive and other classes. Patri gives a criterion: an ℒ∞-space with an infinite-dimensional Schur quotient admits c₀ as a quotient. Its counterexample concerns a missing coordinate embedding needed by a particular explicit map; it does not refute Mazur’s universal statement.[1] [2]

Countable approximation lets retained vectors be studied through sequences. The problem is equivalent to asking for a quotient with a Schauder basis, where each vector has a convergent expansion in one sequence of basis vectors. This would bring that sequential description to an infinite-dimensional linear image even when the original space has no countable dense list.[1] [2]

The survey recasts the question as whether the original space has a strictly increasing sequence of closed subspaces whose union is dense. This connects a countably describable image to a staged description of the original space: each vector can be approached from this hierarchy, although no stage is the whole space. The quotient question thus links information retained by a continuous linear map to approximation within spaces that may have uncountably many directions.[1] [2]