  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unpublished result of Henson.
