Smooth surjections and surjective restrictions

Given a surjective mapping f: E → 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 C1-smooth surjection whose set of critical values is countable. Finally we show that, when f takes values in the Euclidean space Rn, in order to obtain this result it is not sufficient to assume that the set of critical values of f has zero-measure.