Abstract

The role played by real-valued functions in functional analysis is fundamental. One often considers metrics, or seminorms, or linear functionals, to mention some important examples. We introduce the notion of definable real-valued function in functional analysis: a real-valued function $f$ defined on a structure of functional analysis is definable if it can be "approximated" by formulas which do not involve $f$. We characterize definability of real-valued functions in terms of a purely topological condition which does not involve logic.