We attempt to lay the
groundwork for applying the recently-developed theory of models for the
infinitary languages Lπ𝜖t to analysis. It will be shown that within one of
these languages, axioms may be written whose class of models is precisely
the metric spaces. We show that two complete separable metric spaces are
elementarily equivalent in this language if and only if they are isomorphic and
obtain an elimination of quantifiers for such spaces. A method is developed
for transferring results on metric spaces to structures with metrics whose
relations are closed under the metric topology. This class includes Banach
Spaces.
