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International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 33, Pages 1747-1755

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

Division of Natural Science and Mathematics, Indiana University East, Richmond 47358, IN, USA

Received 6 November 2002

Copyright © 2004 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 g1,gnE. The function τ:E×n, defined by letting τ(x,a1,,an)=x+i=1naigi for all xE and all a1,,an, defines an n-type over E. Types that can be represented in this way are called double-dual n-types; we say that (g1,gn)(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.