International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2004 / Article

Open Access

Volume 2004 |Article ID 951346 | https://doi.org/10.1155/S0161171204211152

Markus Pomper, "Double-dual n-types over Banach spaces not containing 1", International Journal of Mathematics and Mathematical Sciences, vol. 2004, Article ID 951346, 9 pages, 2004. https://doi.org/10.1155/S0161171204211152

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

Received06 Nov 2002

Abstract

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.

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.


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