Extraction of critical points of smooth functions on Banach spaces

Abstract

Let $E$ be an infinite-dimensional separable Hilbert space. We show that for every $C^1$ function $f:E\to\mathbb{R}^d$, every open set $U$ with $C_f:=\{x\in E:\,Df(x)\; \text{is not surjective}\}\subset U$ and every continuous function $\varepsilon:E\to (0,\infty)$ there exists a $C^1$ mapping $\varphi:E\to\mathbb{R}^d$ such that $||f(x)-\varphi(x)||\leq \varepsilon(x)$ for every $x\in E$, $f=\varphi$ outside $U$ and $\varphi$ has no critical points ($C_{\varphi}=\emptyset$). This result can be generalized to the case where $E=c_0$ or $E=l_p$, $1<p<\infty$. In the case $E=c_0$ it is also possible to get that $||Df(x)-D\varphi(x)||\leq\varepsilon(x)$ for every $x\in E$.