Double-dual n-types over Banach spaces not containing ℓ1

Let E be a Banach space. The concept of n-type over E is introduced here, generalizing the concept of type over E introduced by Krivine and Maurey. Let E′′ be the second dual of E and fix g′′ 1 , . . . ,g′′ n ∈ E′′. The function τ : E×Rn → R, defined by letting τ(x,a1, . . . ,an) = ‖x+∑ni=1aig′′ i ‖ for all x ∈ E and all a1, . . . ,an ∈R, defines an n-type over E. Types that can be represented in this way are called double-dual n-types; we say that (g′′ 1 , . . . ,g′′ n)∈ (E′′)n realizes τ . Let E be a (not necessarily separable) Banach space that does not contain 1. We study the set of elements of (E′′)n that realize a given double-dual n-type over E. We show that the set of realizations of this n-type is convex. This generalizes a result of Haydon and Maurey who showed that the set of realizations of a given 1-type over a separable Banach space E is convex. The proof makes use of Henson’s language for normed space structures and uses ideas from mathematical logic, most notably the Löwenheim-Skolem theorem.