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Azagra Rueda, Daniel and Jiménez Sevilla, María del Mar and Deville, Robert
(2003)
*On the range of the derivatives of a smooth function between
Banach spacesy.*
Mathematical Proceedings of the Cambridge Philosophical Society, 134
(1).
pp. 163-185.
ISSN 1469-8064

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Official URL: http://journals.cambridge.org/action/displayJournal?jid=PSP

## Abstract

We study the size of the range of the derivatives of a smooth function between Banach spaces. We establish conditions on a pair of Banach spaces X and Y to ensure the existence of a C-p smooth (Frechet smooth or a continuous G (a) over cap teaux smooth) function f from X onto Y such that f vanishes outside a bounded set and all the derivatives of f are surjections. In particular we deduce the following results. For the Gateaux case, when X and Y are separable and X is infinite-dimensional, there exists a continuous G (a) over cap teaux smooth function f from X to Y, with bounded support, so that f'(X) = L (X, Y). In the Frechet case, we get that if a Banach space X has a Frechet smooth bump and dens X = dens L(X, Y), then there is a Frechet smooth function f: X --> Y with bounded support so that f'(X) = L(X, Y). Moreover, we see that if X has a C-p smooth bump with bounded derivatives and dens X = dens L-s(m) (X; Y) then there exists another C-p smooth function f : X --> Y so that f((k)) (X) = L-s(k) (X; Y) for all k = 0,1,...,m. As an application, we show that every bounded starlike body on a separable Banach space X with a (Frechet or G (a) over cap teaux) smooth bump can be uniformly approximated by smooth bounded starlike bodies whose cones of tangent hyperplanes fill the dual space X-*. In the non-separable case, we prove that X has such property if X has smooth partitions of unity.

Item Type: | Article |
---|---|

Uncontrolled Keywords: | Smooth bump; Smooth surjection; Banach space; Smooth body; Derivative |

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

ID Code: | 13961 |

Deposited On: | 25 Nov 2011 12:28 |

Last Modified: | 06 Feb 2014 09:56 |

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