Banach spaces from a model-theoretic viewpoint

Stefano B.


We deal with first-order structures based on Banach spaces.
Henson showed that the ``correct'' logic for Banach spaces is not
first-order logic, but
something such as the {\em positive bounded formulas} with {\em
approximate
satisfiability}.
In a different setting, Fajardo and Keisler formulated an abstract
framework in which
techniques from nonstandard analysis can be applied. As related
development, Keisler
defined and studied a class
of infinitary expressions called {\em neocompact formulas} for which he
proved general
results on
quantifier elimination in {\em law structures}.
We investigate the problem of which Banach spaces have the property
that
neocompact formulas reduce
to countable
conjunctions of quantifier-free positive bounded formulas (notice that
positive bounded
formulas are neocompact). We refer to this property as {\em quantifier
elimination} ({\em
QE}\/). We provide examples of Banach spaces that have {\em QE} and we
find sufficient
conditions under which {\em QE} transfers from nonstandard hull to the
original
space. We also compare {\em QE} with a different notion of quantifier
elimination due to
Henson.