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 ? < c be a cardinal of uncountable cofinality. It is consistent that the continuum be arbitrary large, no Banach space X of density ?, ? < ? < c, condenses onto a compact metric space, but any Banach space of density ? admits a condensation onto a compact metric space. In particular, for ? = ?1, it is consistent that c is arbitrarily large, no Banach space of density ?, ?1 < ? < 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 comact metric space?