Smooth surjections and surjective restrictions
Copyright policy )Smooth surjections and surjective restrictions
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.
11 pages, a small mistake in the proof of Theorem 8 has been fixed
- University of Pisa Italy
- Complutense University of Madrid Spain
- University of Jyväskylä Finland
Topología, 515.1, 46B80, 46T20, smooth surjective mapping, surjective restriction, nonlinear quotient, Uniformly open map, Continuous and differentiable maps in nonlinear functional analysis, 517.98, uniformly open map, Surjective restriction, Mathematics - Metric Geometry, FOS: Mathematics, Matematiikka, Selections in general topology, Special maps on metric spaces, Density character; Nonlinear quotient; Smooth surjective mapping; Surjective restriction; Uniformly open map; Mathematics (all), Análisis funcional y teoría de operadores, ta111, Metric Geometry (math.MG), 1210 Topología, Density character, metriset avaruudet, Functional Analysis (math.FA), Mathematics - Functional Analysis, Nonlinear quotient, density character, Nonlinear classification of Banach spaces; nonlinear quotients, Smooth surjective mapping, funktionaalianalyysi, Mathematics
Topología, 515.1, 46B80, 46T20, smooth surjective mapping, surjective restriction, nonlinear quotient, Uniformly open map, Continuous and differentiable maps in nonlinear functional analysis, 517.98, uniformly open map, Surjective restriction, Mathematics - Metric Geometry, FOS: Mathematics, Matematiikka, Selections in general topology, Special maps on metric spaces, Density character; Nonlinear quotient; Smooth surjective mapping; Surjective restriction; Uniformly open map; Mathematics (all), Análisis funcional y teoría de operadores, ta111, Metric Geometry (math.MG), 1210 Topología, Density character, metriset avaruudet, Functional Analysis (math.FA), Mathematics - Functional Analysis, Nonlinear quotient, density character, Nonlinear classification of Banach spaces; nonlinear quotients, Smooth surjective mapping, funktionaalianalyysi, Mathematics
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