  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 $X$.
