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Smooth surjections and surjective restrictions (1607.01725v2)
Published 6 Jul 2016 in math.FA and math.MG
Abstract: 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 $C1$-smooth surjection whose set of critical values is countable. Finally we show that, when $f$ takes values in the Euclidean space $\mathbb Rn$, in order to obtain this result it is not sufficient to assume that the set of critical values of $f$ has zero-measure.