Note on the Banach Problem 1 of condensations of Banach spaces onto compacta

Abstract

It is consistent with any possible value of the continuum $\mathfrak{c}$ that every infinite-dimensional Banach space of density $\leq \mathfrak{c}$ condenses onto the Hilbert cube. Let $\mu$ be a cardinal of uncountable cofinality. It is consistent that the continuum be arbitrary large, no Banach space $X$ of density $\gamma$, $\mu< \gamma < \mathfrak{c}$ condenses onto a compactum, but any Banach space of density $\mu$ admit a condensation onto a compactum. In particular, for $\mu=\omega_1$, it is consistent that $\mathfrak{c}$ is arbitrarily large, no Banach space of density $\gamma$, $\omega_1< \gamma < \mathfrak{c}$, condenses onto a compactum. These results imply a complete answer to the Problem 1 in the Scottish Book for Banach spaces: When does a Banach space X admit a bijective continuous mapping onto a compact metric space?