This is an announcement for the paper "Approximation by smooth functions with no critical points on separable Banach spaces" by D. Azagra and M. Jimenez-Sevilla.

Abstract: 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 $\varepsilon:X\to\mathbb(0,+\infty)$ there exists a $C^1$ smooth function $g:X\to\mathbb{R}$ for which $|f(x)-g(x)|\leq\varepsilon(x)$ 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.

Archive classification: Functional Analysis; Differential Geometry

Mathematics Subject Classification: 46B20; 46T30; 58E05; 58C25

Remarks: 34 pages

The source file(s), critical270905.tex: 127379 bytes, separable2argument.eps: 46690 bytes, separable3argument.eps: 48762 bytes, separable3bargument.eps: 48459 bytes, sinbase2.eps: 47562 bytes, is(are) stored in gzipped form as 0510603.tar.gz with size 90kb. The corresponding postcript file has gzipped size 220kb.

Submitted from: daniel_azagra@mat.ucm.es

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