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