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15 July 2004 Uniform approximation of continuous mappings by smooth mappings with no critical points on Hilbert manifolds
Daniel Azagra, Manuel Cepedello Boiso
Duke Math. J. 124(1): 47-66 (15 July 2004). DOI: 10.1215/S0012-7094-04-12412-1

Abstract

We prove that every continuous mapping from a separable infinite-dimensional Hilbert space X into $\mathbb{R}^{m}$ can be uniformly approximated by C-smooth mappings with no critical points. This kind of result can be regarded as a sort of 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 that is a level set of a smooth function with no critical points. 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 modeled on X.

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Daniel Azagra. Manuel Cepedello Boiso. "Uniform approximation of continuous mappings by smooth mappings with no critical points on Hilbert manifolds." Duke Math. J. 124 (1) 47 - 66, 15 July 2004. https://doi.org/10.1215/S0012-7094-04-12412-1

Information

Published: 15 July 2004
First available in Project Euclid: 30 July 2004

zbMATH: 1060.57015
MathSciNet: MR2072211
Digital Object Identifier: 10.1215/S0012-7094-04-12412-1

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Rights: Copyright © 2004 Duke University Press

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Vol.124 • No. 1 • 15 July 2004
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