2010/05/19 by Christina Brech, Brech, Christina, Piotr Koszmider +1 · 1 citation
Mathematics · #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA)
paper · pdf · doi:10.48550/arxiv.1005.3530
openalex publication_date 2010/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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 \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 ℓ_∞/c0 is not among them. This relies on the proof of the consistency of the nonexistence of an isomorphic embedding of C([0,\cc]) into ℓ_∞/c0.