A topological group observation on the Banach-Mazur separable quotient problem (Q2415933)

The authors show that every infinite-dimensional Banach space has \(\mathbb{T}^\omega\) as a quotient group, where \(\mathbb{T}\) denotes the compact unit circle group. Indeed, they prove, in Theorem 2.1, the same result in a more general setting. In detail, if \(E\) is a locally convex space (over either the real or the complex field) which contains as a subspace an infinite-dimensional Fréchet space, then \(E\) has \(\mathbb{T}^\omega\) as a quotient group. In addition, the authors provide two examples: \begin{itemize} \item Example 2.4 shows that the condition in Theorem 2.1 is not necessary. \item Example 2.5 shows that not every complete locally convex space has \(\mathbb{T}^\omega\) as a quotient group. \end{itemize}