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

Abstract

It is consistent with any possible value of the continuum c that every infinite-dimensional Banach space of density ≤ c condenses onto the Hilbert cube. Let µ < c be a cardinal of uncountable cofinality. It is consistent that the continuum be arbitrary large, no Banach space X of density γ, µ < γ < c condenses onto a compact metric space, but any Banach space of density µ admit a condensation onto a compact metric space. In particular, for µ = ω1, it is consistent that c is arbitrarily large, no Banach space of density γ, ω1 < γ < c, condenses onto a compact metric space. 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?