The separable quotient problem for barrelled spaces

Abstract

In this survey, we discuss the classical open problem: Does every infinite-dimensional Banach space admit a quotient (by a closed subspace) which is infinite-dimensional and separable? We furnish several equivalent formulations of this famous unsolved problem, in terms of certain Baire-type covering and barrel properties of locally convex spaces, and also in terms of the structure theory of (metrizable, normable) (LF)-spaces. We solve the corresponding Quotient for the class of (LF)-spaces in the affirmative, by actually constructing the separable quotient. Based on the Baire-type covering properties, all (LF)-spaces are partitioned into three mutually disjoint and sufficiently rich classes, called (LFh, (LFh and (LFh-spaces. These three classes are then characterized in terms of the sequence space <po We show that every (LFh-space admits a separable quotient, that is a Frechet space. Every (LFh and (LFh space (more generally, all non-strict (LF)-spaces) possesses a defining sequence, each of whose members has a separable quotient. The intimate interaction between the Quotient Problem for Banach spaces, and the existence of metrizable, as well as normable (LF)-spaces will be studied, resulting in a rich supply of metrizable, as well as normable (LF)-spaces. Finally, we discuss Properly Separable quotients in the setting of barrelled spaces. AMS Subject Classification: 46 A