  We introduce and study an axiomatic theory of $V$-normed $U$-modules, where
$V$ is a Riesz space and $U$ is an $f$-algebra; the spaces $U$ and $V$ also
have some additional structure and are required to satisfy a compatibility
condition. Roughly speaking, a $V$-normed $U$-module is a module over $U$ that
is endowed with a pointwise norm operator taking values in $V$. The aim of our
approach is to develop a unified framework, which is tailored to the
differential calculus on metric measure spaces, where as $U$ and $V$ one can
take many different spaces of functions.
