Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Extraction of critical points of smooth functions on Banach spaces   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 $x\in E$.

14
The categories of this paper are: math-ph
7
The categories of this paper are: hep-th
15
The categories of this paper are: nucl-ex
28
The categories of this paper are: alg-geom
25
The categories of this paper are: patt-sol
9
The categories of this paper are: nlin
30
The categories of this paper are: q-alg
20
The categories of this paper are: acc-phys
31
The categories of this paper are: ao-sci
23
The categories of this paper are: solv-int
12
The categories of this paper are: nucl-th
6
The categories of this paper are: astro-ph
16
The categories of this paper are: stat
19
The categories of this paper are: eess
33
The categories of this paper are: plasm-ph
24
The categories of this paper are: dg-ga