Contributions to model theory of metric structures

Continuous logic;Metric structures;Model theory;Perturbations;Banach spaces;Algebraic closure

Tellez, Hernando

关键词: Continuous logic;    Metric structures;    Model theory;    Perturbations;    Banach spaces;    Algebraic closure;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/16109/1_Tellez_Hernando.pdf?sequence=2&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

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.