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2005 Definable Types Over Banach Spaces
José Iovino
Notre Dame J. Formal Logic 46(1): 19-50 (2005). DOI: 10.1305/ndjfl/1107220672

Abstract

We study connections between asymptotic structure in a Banach space and model theoretic properties of the space. We show that, in an asymptotic sense, a sequence $(x_n)$ in a Banach space X generates copies of one of the classical sequence spaces $\ell_p$ or $c_0$ inside X (almost isometrically) if and only if the quantifier-free types approximated by $(x_n)$ inside X are quantifier-free definable. More precisely, if $(x_n)$ is a bounded sequence X such that no normalized sequence of blocks of $(x_n)$ converges, then the following two conditions are equivalent. (1) There exists a sequence $(y_n)$ of blocks of $(x_n)$ such that for every finite dimensional subspace E of X, every quantifier-free type over $E +\overline{\rm span}\{y_n\mid n\in \mathbb{N}\}$ is quantifier-free definable. (2) One of the following two conditions holds: (a) there exists $1\le p< \infty$ such that for every $\epsilon>0$ and every finite dimensional subspace E of X there exists a sequence of blocks of $(x_n)$ which is $(1+\epsilon)$equivalent over E to the standard unit basis of $\ell_p$; (b) for every $\epsilon>0$ and every finite dimensional subspace E of X there exists a sequence of blocks of $(x_n)$ which is $(1+\epsilon)$-equivalent over E to the standard unit basis of $c_0$. Several byproducts of the proof are analyzed.

Citation

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José Iovino. "Definable Types Over Banach Spaces." Notre Dame J. Formal Logic 46 (1) 19 - 50, 2005. https://doi.org/10.1305/ndjfl/1107220672

Information

Published: 2005
First available in Project Euclid: 31 January 2005

zbMATH: 1082.46010
MathSciNet: MR2131545
Digital Object Identifier: 10.1305/ndjfl/1107220672

Subjects:
Primary: 03C
Secondary: 46B

Rights: Copyright © 2005 University of Notre Dame

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Vol.46 • No. 1 • 2005
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