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It is consistent with any possible value of the continuum $\mathfrak{c}$ thatevery infinite-dimensional Banach space of density $\leq \mathfrak{c}$condenses onto the Hilbert cube. Let $\mu$ be a cardinal of uncountablecofinality. It is consistent that the continuum be arbitrary large, no Banachspace $X$ of density $\gamma$, $\mu< \gamma < \mathfrak{c}$ condenses onto acompactum, but any Banach space of density $\mu$ admit a condensation onto acompactum. 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 resultsimply a complete answer to the Problem 1 in the Scottish Book for Banachspaces: When does a Banach space X admit a bijective continuous mapping onto acompact metric space?

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