The quotient/codimension problems

Abstract

If a Frechet space admits a separable quotient, so do its countable-codimensional subspaces. We generalize to metrizable primitive spaces, function spaces, most \(\left( LF\right) \)-spaces, many others. New characterizations emerge: (1) A locally convex space E is primitive if and only if, given any closed countable-codimensional subspace F and dense subspace D in E, the intersection \(F\bigcap D\) is dense in F; (2) A completely regular Hausdorff space X is pseudocompact if and only if every infinite-dimensional subspace of \(L_{p}\left( X\right) \) admits a quotient that is properly separable in the sense of Robertson.