A topological group observation on the Banach–Mazur separable quotient problem

Abstract

The Separable Quotient Problem of Banach and Mazur asks if every infinite-dimensional Banach space has an infinite-dimensional separable quotient Banach space. It has remained unsolved for 85 years but has been answered in the affirmative for special cases such as reflexive Banach spaces. An affirmative answer to the Separable Quotient Problem would obviously imply that every infinite-dimensional Banach space has a quotient topological group which is separable, metrizable, and infinite-dimensional in the sense of topology. In this paper it is proved that every infinite-dimensional Banach space has as a quotient group the separable metrizable infinite-dimensional topological group, Tω, where T denotes the compact unit circle group. Indeed it is shown that every locally convex space, which has a subspace which is an infinite-dimensional Fréchet space, has Tω as a quotient group.