International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 33, Pages 1747-1755
Double-dual -types over Banach spaces not containing
Division of Natural Science and Mathematics, Indiana University East, Richmond 47358, IN, USA
Received 6 November 2002
Copyright © 2004 Markus Pomper. 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 be a Banach space. The concept of -type over is introduced here, generalizing the concept of type over introduced by Krivine and Maurey. Let be the second dual of and fix . The function , defined by letting for all and all , defines an -type over . Types that can be represented in this way are called double-dual -types; we say that realizes . Let be a (not necessarily separable) Banach space that does not contain . We study the set of elements of that realize a given double-dual -type over . We show that the set of realizations of this -type is convex. This generalizes a result of Haydon and Maurey who showed that the set of realizations of a given -type over a separable Banach space 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.