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RESEARCH PRODUCT
Smooth surjections and surjective restrictions
Richard M. AronJesús A. JaramilloEnrico Le Donnesubject
TopologíaPure mathematicsmetric spaces46B80 46T20General Mathematicssmooth surjective mappingBanach spacesurjective restrictionnonlinear quotient01 natural sciencesfunctional analysisSurjective functionuniformly open mapMathematics - Metric GeometryFOS: MathematicsMathematics (all)Order (group theory)Countable set0101 mathematicsAnálisis funcional y teoría de operadoresDensity character; Nonlinear quotient; Smooth surjective mapping; Surjective restriction; Uniformly open map; Mathematics (all)MathematicsEuclidean spaceta111010102 general mathematicsMetric Geometry (math.MG)16. Peace & justicemetriset avaruudetFunctional Analysis (math.FA)Mathematics - Functional Analysis010101 applied mathematicsCharacter (mathematics)density characterfunktionaalianalyysiBijection injection and surjectionSubspace topologydescription
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.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2017-01-01 |