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Summary: UNIFORM APPROXIMATION OF CONTINUOUS
MAPPINGS BY SMOOTH MAPPINGS WITH NO
CRITICAL POINTS ON HILBERT MANIFOLDS
DANIEL AZAGRA and MANUEL CEPEDELLO BOISO
Abstract
We prove that every continuous mapping from a separable infinite-dimensional
Hilbert space X into Rm can be uniformly approximated by C-smooth mappings
with no critical points. This kind of result can be regarded as a sort of strong ap-
proximate version of the Morse-Sard theorem. Some consequences of the main the-
orem are as follows. Every 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-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 modeled on X.
1. Introduction and main results
A fundamental result in differential topology and analysis is the Morse-Sard theorem
(see [21], [22]), which states that if f : Rn - Rm is a Cr -smooth function with
r > max{n-m, 0} and C f stands for the set of critical points of f (i.e., the points x at
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