Uniform approximation of continuous mappings by smooth mappings with no critical points on Hilbert manifolds
| dc.creator | Azagra, Daniel | |
| dc.creator | Boiso, Manuel Cepedello | |
| dc.date | 2002-03-22 | |
| dc.date.accessioned | 2026-07-07T04:47:14Z | |
| dc.date.available | 2026-07-07T04:47:14Z | |
| dc.description | We prove that every continuous mapping from a separable infinite-dimensional Hilbert space $X$ into $\mathbb{R}^{m}$ can be uniformly approximated by $C^\infty$ smooth mappings {\em 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 $C^\infty$ 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 $X$. | |
| dc.description | 23 pages, improved version of a previous preprint | |
| dc.identifier | https://arxiv.org/abs/math/0203237 | |
| dc.identifier | http://arxiv.org/abs/math/0203237 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/63636 | |
| dc.subject | Differential Geometry | |
| dc.subject | Functional Analysis | |
| dc.subject | 58A05, 58B99, 57R12, 46T05 | |
| dc.title | Uniform approximation of continuous mappings by smooth mappings with no critical points on Hilbert manifolds | |
| dc.type | text |