  Given a surjective mapping $f : E \to F$ between Banach spaces, we
investigate the existence of a subspace $G$ of $E$, with the same density
character as $F$, such that the restriction of $f$ to $G$ remains surjective.
We obtain a positive answer whenever $f$ is continuous and uniformly open. In
the smooth case, we deduce a positive answer when $f$ is a $C^1$-smooth
surjection whose set of critical values is countable. Finally we show that,
when $f$ takes values in the Euclidean space $\mathbb R^n$, in order to obtain
this result it is not sufficient to assume that the set of critical values of
$f$ has zero-measure.
