On universal Banach spaces of density continuum

TL;DR

This paper investigates the existence of a universal Banach space of density continuum, showing it is independent of standard set-theoretic axioms and exploring conditions under which such spaces exist or do not.

Contribution

It proves the consistency of the nonexistence of a universal Banach space of density continuum within standard set theory, and explores the variability of such spaces.

Findings

Existence of a universal Banach space of density continuum is independent of ZFC.

Under certain set-theoretic assumptions, no universal Banach space of density continuum exists.

It is consistent that universal Banach spaces exist but do not include 

Abstract

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 $\ell_\infty/c_0$.