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Uniform approximation of continuous mappings by smooth mappings with no critical points on Hilbert manifolds. (English) Zbl 1060.57015

Authors’ summary: 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 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. Any 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 \(C^\infty\)-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 \(M\) modified on \(X\).

MSC:

57R12 Smooth approximations in differential topology
57R45 Singularities of differentiable mappings in differential topology
58B10 Differentiability questions for infinite-dimensional manifolds
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