#
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.