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 C-1-smooth surjection whose set of critical values is countable. Finally we show that, when f takes values in the Euclidean space 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.
Ministerio de Economía y Competitividad (MINECO)
Academy of Finland
Depto. de Análisis Matemático y Matemática Aplicada
Fac. de Ciencias Matemáticas
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