Double-dual n -types over Banach spaces not containing ℓ 1

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 × ℝ n → ℝ , defined by letting τ ( x , a 1 , … , a n ) = ‖ x + ∑ i = 1 n a i g ″ i ‖ for all x ∈ E and all a 1 , … , a n ∈ 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 ℓ 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.