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Let $E$, $F$ be separable Hilbert spaces, and assume that $E$ isinfinite-dimensional. We show that for every continuous mapping $f:E\to F$ andevery 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 alsoprovide a version of this result where $E$ can be replaced with a Banach spacefrom a large class (including all the classical spaces with smooth norms, suchas $c_0$, $\ell_p$ or $L^{p}$, $1<p<\infty$), and $F$ can be taken to be anyBanach 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 beuniformly approximated by smooth open mappings. Part of the proof providesresults of independent interest that improve some known theorems aboutdiffeomorphic extractions of closed sets from Banach spaces or Hilbertmanifolds.

Smooth approximations without critical points of continuous mappings between Banach spaces, and diffeomorphic extractions of sets