ℵ₀‐categorical Banach spaces contain ℓp or c₀

This paper has three parts. First, we establish some of the basic model theoretic facts about , the Tsirelson space of Figiel and Johnson [20]. 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) has the non independence property (NIP); (2) every Banach space that is ℵ 0 ‐categorical up to small perturbations embeds c 0 or () almost isometrically; consequently the (continuous) first‐order theory of does not characterize , up to almost isometric isomorphism.