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

Abstract

The Banach-Mazur problem, which asks if every infinite-dimensional Banach space has an infinite-dimensional separable quotient space, has remained unsolved for 85 years, but has been answered in the affirmative for special cases such as reflexive Banach spaces. It is also known that every infinite-dimensional non-normable Fr\'{e}chet space has an infinite-dimensional separable quotient space, namely $\mathbb{R}^\omega $. It is proved in this paper that every infinite-dimensional Fr\'{e}chet space (including every infinite-dimensional Banach space), indeed every locally convex space which has a subspace which is an infinite-dimensional Fr\'{e}chet space, has an infinite-dimensional (in the topological sense) separable metrizable quotient group, namely $\mathbb{T}^\omega$, where $\mathbb{T}$ denotes the compact unit circle group.