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Uniform approximation of continuous mappings by smooth mappings with no critical points on Hilbert manifolds

Opened Access Uniform approximation of continuous mappings by smooth mappings with no critical points on Hilbert manifolds
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Autor: Azagra Rueda, Daniel
Cepedello Boiso, Manuel
Departamento: Universidad de Sevilla. Departamento de Análisis Matemático
Fecha: 2004-07-15
Publicado en: Duke Mathematical Journal, 124 (1), 47-66.
Tipo de documento: Artículo
Resumen: We prove that every continuous mapping from a separable infinitedimensional Hilbert space X into R m can be uniformly approximated by C∞ smooth mappings with no critical points. This kind of result can be regarded as a sort of very strong approximate version of the Morse-Sard theorem. Some consequences of the main theorem are as follows. Every two disjoint closed subsets of X can be separated by a one-codimensional smooth manifold which is a level set of a smooth function with no critical points; this fact may be viewed as a nonlinear analogue of the geometrical version of the Hahn-Banach theorem. In particular, every closed set in X can be uniformly approximated by open sets whose boundaries are C∞ smooth one-codimensional submanifolds of X. Finally, since every Hilbert manifold is diffeomorphic to an open subset of the Hilbert space, all of these results still hold if one replaces the Hilbert space X with any smooth manifold M modelled on X.
Cita: Azagra Rueda, D. y Cepedello Boiso, M. (2004). Uniform approximation of continuous mappings by smooth mappings with no critical points on Hilbert manifolds. Duke Mathematical Journal, 124 (1), 47-66.
Tamaño: 237.9Kb
Formato: PDF

URI: http://hdl.handle.net/11441/43510

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