Geodesic
Note on the Banach problem 1 of condensations of Banach spaces onto compacta
Filomat, Tome 37 (2023) no. 7, p. 2183

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DOI

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 admits a condensation onto a compact metric space. In particular, for µ = ω 1 , it is consistent that c is arbitrarily large, no Banach space of density γ, ω 1 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?
DOI : 10.2298/FIL2307183O
Classification : 57N20, 54C10, 54E99
Keywords: Banach problem, condensation, metric compact space, density 
Alexander V Osipov. Note on the Banach problem 1 of condensations of Banach spaces onto compacta. Filomat, Tome 37 (2023) no. 7, p. 2183 . doi: 10.2298/FIL2307183O
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     title = {Note on the {Banach} problem 1 of condensations of {Banach} spaces onto compacta},
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     doi = {10.2298/FIL2307183O},
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     url = {http://geodesic.mathdoc.fr/articles/10.2298/FIL2307183O/}
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