Approximation by smooth functions with no critical points on separable Banach spaces
| dc.creator | Azagra, D. | |
| dc.creator | Jimenez-Sevilla, M. | |
| dc.date | 2005-10-27 | |
| dc.date.accessioned | 2026-07-07T06:48:01Z | |
| dc.date.available | 2026-07-07T06:48:01Z | |
| dc.description | We characterize the class of separable Banach spaces $X$ such that for every continuous function $f:X\to\mathbb{R}$ and for every continuous function $ε:X\to\mathbb(0,+\infty)$ there exists a $C^1$ smooth function $g:X\to\mathbb{R}$ for which $|f(x)-g(x)|\leqε(x)$ and $g'(x)\neq 0$ for all $x\in X$ (that is, $g$ has no critical points), as those Banach spaces $X$ with separable dual $X^*$. We also state sufficient conditions on a separable Banach space so that the function $g$ can be taken to be of class $C^p$, for $p=1,2,..., +\infty$. In particular, we obtain the optimal order of smoothness of the approximating functions with no critical points on the classical spaces $\ell_p(\mathbb{N})$ and $L_p(\mathbb{R}^n)$. Some important consequences of the above results are (1) the existence of {\em a non-linear Hahn-Banach theorem} and (2) the smooth approximation of closed sets, on the classes of spaces considered above. | |
| dc.description | 34 pages | |
| dc.identifier | https://arxiv.org/abs/math/0510603 | |
| dc.identifier | http://arxiv.org/abs/math/0510603 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/103844 | |
| dc.subject | Functional Analysis | |
| dc.subject | Differential Geometry | |
| dc.subject | 46B20; 46T30; 58E05; 58C25 | |
| dc.title | Approximation by smooth functions with no critical points on separable Banach spaces | |
| dc.type | text |