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Paper IPM / M / 16722  


Abstract:  
This paper has three parts. First, we establish some of the basic model theoretic facts about T, the Tsirelson space of Figiel and Johnson [FJ74]. 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) every strongly separable space, such as MT , has the non independence property (NIP); (2) every
@0categorical space is stable if and only if it is strongly separable; consequently T is not alephcategorical; equivalently, its firstorder theory (in any countable language) does not characterize T, up to isometric isomorphism; (3) every explicitly definable space contains isometric copies of c_0 or `_p.
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