Calculus of Variations and Geometric Measure Theory
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Richard M. Aron - Jesús A. Jaramillo - E. Le Donne

Smooth surjections and surjective restrictions

created by ledonne on 21 Dec 2018



Inserted: 21 dec 2018

Year: 2016

ArXiv: 1607.01725 PDF


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.

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