Let E be a Banach space. The concept of n-type overE is introduced here, generalizing the concept of type overE 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+∑i=1naig″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 ℓ1. 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.
