arXiv math/05106032005-10-27math.FAmath.DG

可分巴拿赫空间上无临界点的光滑函数逼近

Approximation by smooth functions with no critical points on separable Banach spaces

D. Azagra, M. Jimenez-Sevilla

备注：: 34 pages

摘要：

我们刻画了可分Banach空间类$X$，使得对于每个连续函数$f:X\to\mathbb{R}$和$ε:X\to\mathbb(0,+\infty)$，存在一个$C^1$光滑函数$g:X\to\mathbb{R}$，对于所有的$x\in X$，$|f(x)-g(x)|\leqε(x)$和$g'(x)\neq 0$（即$g$没有临界点），如同那些具有可分对偶$X^*$的巴拿赫空间$X$.我们还陈述了可分Banach空间上的充分条件，使得函数$g$可以被看作是$C^p$类，对于$p=1,2,..., +\infty$。特别地，我们得到了经典空间$\ell_p(\mathbb{N})$和$L_p(\mathbb{R}^n)$上无临界点的逼近函数的最佳光滑阶.上述结果的一些重要结果是（1）{ \em a non-linear Hahn-Banach theorem}的存在性和（2）闭集在上述考虑的空间类上的光滑逼近。

英文摘要：

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.

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摘要：

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