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Contributions to model theory of metric structures

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Title: Contributions to model theory of metric structures
Author(s): Tellez, Hernando
Director of Research: Henson, Ward
Doctoral Committee Chair(s): Solecki, Slawomir
Doctoral Committee Member(s): Henson, Ward; van den Dries, Lou; Rosendal, Christian
Department / Program: Mathematics
Discipline: Mathematics
Degree Granting Institution: University of Illinois at Urbana-Champaign
Degree: Ph.D.
Genre: Dissertation
Subject(s): Continuous logic Metric structures Model theory Perturbations Banach spaces Algebraic closure
Abstract: Two Banach spaces X and Y are said to be almost isometric if for every λ > 1 there exists a λ-isomorphism f : X → Y . That is, a linear surjective map such that 1/λ ∥x∥ ≤ ∥f (x)∥ ≤ λ ∥x∥ for every x ∈ X . In this thesis we prove a Ryll-Nardzewski-style characterization of ω-categoricity up to almost isometry for Banach spaces using the concept of perturbations of metric structures and tools developed by Ben Yaacov ([6] and [5]). To this end we construct a single-sorted signature Lc for the study of the model theory of Banach spaces in the setting of continuous first order logic, we give an explicit axiomatization for the class of Lc -structures that come from unit balls of Banach spaces and we construct a perturbation system that is adequate for the study of almost isometric Banach spaces. Additionally, we study the algebraic closure construction for metric structures in the setting of continuous first order logic. We give several characterizations of algebraicity, and we prove basic properties analogous to ones that the algebraic closure satisfes in classical first order logic.
Issue Date: 2010-05-19
URI: http://hdl.handle.net/2142/16109
Rights Information: Copyright 2010 Hernando Tellez
Date Available in IDEALS: 2010-05-19
Date Deposited: 2010-05
 

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