Notre Dame Journal of Formal Logic

Definable Types Over Banach Spaces

José Iovino

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.

Article information

Source
Notre Dame J. Formal Logic Volume 46, Number 1 (2005), 19-50.

Dates
First available: 31 January 2005

Permanent link to this document
http://projecteuclid.org/euclid.ndjfl/1107220672

Digital Object Identifier
doi:10.1305/ndjfl/1107220672

Mathematical Reviews number (MathSciNet)
MR2131545

Zentralblatt MATH identifier
02186749

Subjects
Primary: 03C
Secondary: 46B

Keywords
Banach space definable type classical sequence space

Citation

Iovino, José. Definable Types Over Banach Spaces. Notre Dame Journal of Formal Logic 46 (2005), no. 1, 19--50. doi:10.1305/ndjfl/1107220672. http://projecteuclid.org/euclid.ndjfl/1107220672.


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