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

 

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Opened Access Uniform approximation of continuous functions by smooth functions with no critical points on Hilbert manifolds
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Author: Azagra Rueda, Daniel
Cepedello Boiso, Manuel
Department: Universidad de Sevilla. Departamento de Análisis Matemático
Date: 2001
Published in: Conference on Function Theory on Infinite Dimensional Banach Spaces (2001), p 1-24
Document type: Presentation
Abstract: We prove that every continuous function on a separable infinitedimensional Hilbert space X can be uniformly approximated by C∞ smooth functions 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.
Cite: Azagra Rueda, D. y Cepedello Boiso, M. (2001). Uniform approximation of continuous functions by smooth functions with no critical points on Hilbert manifolds. Universidad de Sevilla. FQM260: Variable Compleja y Teoría de Operadores.
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URI: http://hdl.handle.net/11441/54777

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