  Let $E$, $F$ be separable Hilbert spaces, and assume that $E$ is
infinite-dimensional. We show that for every continuous mapping $f:E\to F$ and
every continuous function $\varepsilon: E\to (0, \infty)$ there exists a
$C^{\infty}$ mapping $g:E\to F$ such that $\|f(x)-g(x)\|\leq\varepsilon(x)$ and
$Dg(x):E\to F$ is a surjective linear operator for every $x\in E$. We also
provide a version of this result where $E$ can be replaced with a Banach space
from a large class (including all the classical spaces with smooth norms, such
as $c_0$, $\ell_p$ or $L^{p}$, $1<p<\infty$), and $F$ can be taken to be any
Banach space such that there exists a bounded linear operator from $E$ onto
$F$. In particular, for such $E, F$, every continuous mapping $f:E\to F$ can be
uniformly approximated by smooth open mappings. Part of the proof provides
results of independent interest that improve some known theorems about
diffeomorphic extractions of closed sets from Banach spaces or Hilbert
manifolds.
