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 $\epsilon :E\to (0,\infty )$ there exists a $C^1$ mapping $\phi :E\to {\mathbb{R}}^d$ such that $||f(x)-\phi (x)||\le \epsilon (x)$ for every $x\in E$, $f=\phi$ outside $U$ and $\phi$ has no critical points ($C_{\phi }=\varnothing$). 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 $||Df(x)-D\phi (x)||\le \epsilon (x)$ for every $x\in E$.

arXiv comment20 pages; The introduction has been rewritten. An errata concerning some estimates with the unconditional constant has been corrected