Double-dual \(n\)-types over Banach spaces not containing \(\ell_{1}\)
Summary: 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 \(g_1'', \dotsc, g_n'' \in E''\). The function \(\tau : E \times \mathbb{R}^n \rightarrow \mathbb{R}\), defined by letting \(\tau(x, a_1, \dotsc, a_n) =\|x + \sum_{i=1}^n a_i g_i''\|\) for all \(x \in E\) and all \(a_1, \dotsc, a_n \in \mathbb{R}\), defines an \(n\)-type over \(E\). Types that can be represented in this way are called double-dual \(n\)-types; we say that \((g_1'', \dotsc, g_n'') \in(E'')^n\) realizes \(\tau\). Let \(E\) be a (not necessarily separable) Banach space that does not contain \(\ell_{1}\). 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.
scientific article; zbMATH DE number 2168723