Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 42, 2017, 525-534

SMOOTH SURJECTIONS AND SURJECTIVE RESTRICTIONS

Richard M. Aron, Jesús Angel Jaramillo and Enrico Le Donne

Kent State University, Department of Mathematical Sciences
Kent, Ohio 44242, U.S.A.; aron 'at' math.kent.edu

Universidad Complutense de Madrid, Instituto de Matemática Interdiscliplinar (IMI) and Departamento de Análisis Matemático
28040-Madrid, Spain; jaramil 'at' mat.ucm.es

University of Jyväskylä, Department of Mathematics and Statistics
40014 Jyväskylä, Finland; ledonne 'at' msri.org

Abstract. 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 C¹-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.

2010 Mathematics Subject Classification: Primary 46B80, 46T20, 54E40, 54C65.

Key words: Smooth surjective mapping, nonlinear quotient, surjective restriction, uniformly open map, density character.

Reference to this article: R. M. Aron, J. A. Jaramillo and E. Le Donne: Smooth surjections and surjective restrictions. Ann. Acad. Sci. Fenn. Math. 42 (2017), 525-534.

Full document as PDF file

https://doi.org/10.5186/aasfm.2017.4237