### Impacto

Azagra Rueda, Daniel and Jimenez Sevilla, Maria del Mar
(2007)
*Approximation by smooth functions with no critical points on separable Banach spaces.*
Journal of Functional Analysis , 242
(1).
pp. 1-36.
ISSN 0022-1236

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Official URL: http://www.sciencedirect.com/science/article/pii/S0022123606003600

## Abstract

We characterize the class of separable Banach spaces X such that for every continuous function f : X -> Rand for every continuous function epsilon : X -> (0, +infinity) there exists a C-1 smooth function g: X -> R for which vertical bar f(x) - g(x)vertical bar <= epsilon(x) and g'(x) not equal 0 for all x is an element of X (that is, g has no critical points), as those infinite-dimensional 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,..., +infinity. In particular, we obtain the optimal order of smoothness of the approximating functions with no critical points on the classical spaces l(p)(N) and L-p(R-n). Some important consequences of the above results are (1) the existence of a non-linear Hahn-Banach theorem and the smooth approximation of closed sets, on the classes of spaces considered above; and (2) versions of all these results for a wide class of infinite-dimensional Banach manifolds.

Item Type: | Article |
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Uncontrolled Keywords: | Rolles theorem; Singular maps; Hilbert-space; Image size; Manifolds; Morse-Sard theorem; smooth bump functions; critical points; approximation by smooth functions; Sard functions |

Subjects: | Sciences > Mathematics > Functional analysis and Operator theory |

ID Code: | 14770 |

Deposited On: | 17 Apr 2012 11:52 |

Last Modified: | 06 Feb 2014 10:08 |

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