2001/12/03 by Daniel Azagra, Azagra, Daniel, Manuel Cepedello Boiso +1
Computer Science · Mathematics · #46T05 #57R12 #57R70 #58B99 #Advanced Mathematical Modeling in Engineering #Differential Geometry (math.DG) #FOS: Mathematics #Fixed Point Theorems Analysis #Geometric Analysis and Curvature Flows #math.DG #msc:46T05 #msc:57R12 #msc:57R70 #msc:58B99
paper · pdf · doi:10.48550/arxiv.math/0112020
24 pages
arxiv created 2001/12/03 · openalex publication_date 2001/12/03 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
We prove that every continuous function on a separable infinite-dimensional Hilbert space X can be uniformly approximated by 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 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.