Lifting problems for normed spaces (Q259003)

For a cardinal \(d\), denote by \(\ell_0^1(d)\) the space of all finitely supported functions \(f: d \to \mathbb C\), equipped with the norm \(\|f\| = \sum_{t \in d} |f(t)|\). A normed space \(E\) is said to be topologically projective in the category of normed spaces if it has the following lifting property: For every pair of normed spaces \(Y, Z\), for every continuous open linear map \(\psi: Y \to Z\) and every continuous linear map \(\phi: E \to Z\) there is a continuous linear map \(\delta: E \to Y\) such that \(\psi \circ \delta = \phi\). Answering a question by \textit{A. Ya. Helemskiĭ} [private communication], the author demonstrates that a normed space \(E\) is topologically projective in the category of normed spaces if and only if it is isomorphic to \(\ell_0^1(\dim E)\).