On universal Banach spaces of density continuum

الأصل · EN

We consider the question whether there exists a Banach space X of density continuum such that every Banach space of density not bigger than continuum isomorphically embeds into X (called a universal Banach space of density). It is well known that ℓ∞/c₀ is such a space if we assume the continuum hypothesis. However, some additional set-theoretic assumption is needed, as we prove in the main result of this paper that it is consistent with the usual axioms of set-theory that there is no universal Banach space of density. Thus, the problem of the existence of a universal Banach space of density is undecidable using the usual axioms of set-theory. We also prove that it is consistent that there are universal Banach spaces of density, but ℓ∞/c₀ is not among them. This relies on the proof of the consistency of the nonexistence of an isomorphic embedding of C([0,]) into ℓ∞/c₀.