${\aleph }_0$-categorical Banach spaces contain ${\ell }_p$ or $c_0$

Abstract

This paper has three parts. First, we establish some of the basic model theoretic facts about $M_{\mathcal{T}}$, the Tsirelson space of Figiel and Johnson \cite{FJ}. Second, using the results of the first part, we give some facts about general Banach spaces. Third, we study model-theoretic dividing lines in some Banach spaces and their theories. In~particular, we show: (1) $M_{\mathcal{T}}$ has the \emph{non independence property} (NIP); (2) every Banach space that is ${\aleph }_0$-categorical up to small perturbations embeds $c_0$ or ${\ell }_p$ ($1⩽p<\infty$) almost isometrically; consequently the (continuous) first-order theory of $M_{\mathcal{T}}$ does not characterize $M_{\mathcal{T}}$, up to almost isometric isomorphism.

In the new version, many changes have been made: (1) there are some necessary functional analysis background, (2) several results have been improved, (3) there are many rearrangements, (4) this version is probably more readable. There were some incomplete ideas in the previous version that were removed in this version

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• Topometric spaces and perturbations of metric structures
• Modular functionals and perturbations of Nakano spaces
• Remarks on NIP in a model
• Dividing lines in unstable theories and subclasses of Baire 1 functions
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Cited by in corpus (2)

• Remarks on convergence of Morley sequences
• Glivenko-Cantelli classes and NIP formulas