CONTINUOUS FIRST ORDER LOGIC FOR UNBOUNDED METRIC STRUCTURES ITAI BEN YAACOV 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 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 [Benc] 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 This last result is motivated by an unpublished result of Henson. Introduction Continuous first order logic is an extension of classical first order logic, introduced in [BU] as a model theoretic formalism for metric structures. It is convenient to consider that continuous logic also extends C. Ward Henson's logic for Banach space structures (see for example [HI02]), even though this statement is obviously false: continuous first order logic deals exclusively with bounded metric structures, immediately excluding Banach spaces from the picture.
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