On universal Banach spaces of density continuum

We consider the question whether there exists a Banach space X of density continuum such that every Banach space of density at most continuum isomorphically embeds into X (called a universal Banach space of density c). It is well known that a""(a)/c (0) is such a space if we assume the continuum hypothesis. Some additional set-theoretic assumption is indeed 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 c. Thus, the problem of the existence of a universal Banach space of density c is undecidable using the usual axioms of set-theory. We also prove that it is consistent that there are universal Banach spaces of density c, but a""(a)/c (0) is not among them. This relies on the proof of the consistency of the nonexistence of an isomorphic embedding of C([0, c]) into a""(a)/c (0).

FAPESP

FAPESP [2007/08213-2]

Thematic Project FAPESP [2006/02378-7]

Fapesp (thematic project)

Polish Ministry of Science and Higher Education [N N201 386234]

Polish Ministry of Science and Higher Education

Junta de Andalucia

Junta de Andalucia

FEDER

FEDER [P06-FQM-01438]