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

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

2005-10-27

arxiv.org/abs/math/0510603v1

Mathematics

Functional Analysis

34 pages

Scientific paper

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

Affiliated with

Also associated with

No associations

Rating

Approximation by smooth functions with no critical points on separable Banach spaces does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Approximation by smooth functions with no critical points on separable Banach spaces, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Approximation by smooth functions with no critical points on separable Banach spaces will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-322900