Smooth surjections and surjective restrictions
Aron, R. M., Jaramillo, J. A., & Le Donne, E. (2017). Smooth surjections and surjective restrictions. Annales Academiae Scientiarum Fennicae-Mathematica, 42 (2), 525-534. doi:10.5186/aasfm.2017.4237
Published inAnnales Academiae Scientiarum Fennicae-Mathematica
© the Authors & Suomalainen tiedeakatemia, 2017.
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 Rn, in order to obtain this result it is not sufficient to assume that the set of critical values of f has zero-measure.