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 not bigger than continuum isomorphically embeds into \(X\) (called a universal Banach space of density \(\cc\)). It is well known that \(\ell_\infty/c_0\) is such a space i...

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Bibliographic Details
Published in:arXiv.org 2010-05
Main Authors: Brech, Christina, Koszmider, Piotr
Format: Article
Language:English
Subjects:
Axioms
Banach spaces
Density
Online Access:Get full text
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Summary: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 \(\cc\)). It is well known that \(\ell_\infty/c_0\) 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 \(\cc\). Thus, the problem of the existence of a universal Banach space of density \(\cc\) is undecidable using the usual axioms of set-theory. We also prove that it is consistent that there are universal Banach spaces of density \(\cc\), but \(\ell_\infty/c_0\) is not among them. This relies on the proof of the consistency of the nonexistence of an isomorphic embedding of \(C([0,\cc])\) into \(\ell_\infty/c_0\).
ISSN:2331-8422