## 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

### Keywords:

Morse-Sard theorem; Hilbert manifold
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### References:

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