Continuous first order logic for unbounded metric structures

TL;DR

This paper develops a version of continuous first order logic tailored for unbounded metric structures, enabling analysis of Banach spaces beyond the unit ball approach and establishing a characterization of their theories under perturbations.

Contribution

It introduces unbounded continuous first order logic with emboundment, extending the framework to non-unit ball structures and characterizing Banach space theories with perturbations.

Findings

Introduces unbounded continuous logic with emboundment.

Provides a Ryll-Nardzewski style characterization for Banach spaces.

Extends logical analysis to non-unit ball structures.

Abstract

We present an adaptation of continuous first order logic to unbounded metric structures. This has the advantage of being closer in spirit to C. Ward Henson's logic for Banach space structures than the unit ball approach (which has been the common approach so far to Banach space structures in continuous logic), as well as of applying in situations where the unit ball approach does not apply (i.e., when the unit ball is not a definable set). We also introduce the process of single point \emph{emboundment} (closely related to the topological single point compactification), allowing to bring unbounded structures back into the setting of bounded continuous first order logic. Together with results from \cite{BenYaacov:Perturbations} regarding perturbations of bounded metric structures, we prove a Ryll-Nardzewski style characterisation of theories of Banach spaces which are separably categorical up to small perturbation of the norm. This last result is motivated by an unpublished result of Henson.