[L'image de la dérivée d'une fonction bosse différentiable]
Nous étudions l'image de la dérivée d'une fonction bosse Fréchet différentiable. X est un espace de Banach séparable de dimension infinie et Cp-lisse. Tout d'abord nous montrons que tout ouvert connexe de contenant 0 est l'image de la dérivée d'une bosse de classe Cp. Ensuite, les parties analytiques de qui vérifient une condition naturelle de liaison sont l'image de la dérivée d'une bosse de classe C1. Nous trouvons des résultats analogues en dimension finie. Finalement, nous prouvons que si f est une C2-bosse sur , est l'adhérence de son intérieur.
We study the range of the derivative of a Frechet differentiable bump. X is an infinite dimensional separable Cp-smooth Banach space. We first prove that any connected open subset of containing 0 is the range of the derivative of a Cp-bump. Next, analytic subsets of which satisfy a natural linkage condition are the range of the derivative of a C1-bump. We find analogues of these results in finite dimensions. We finally show that is the closure of its interior, if f is a C2-bump on .
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@article{CRMATH_2002__334_3_189_0,
author = {Gaspari, Thierry},
title = {The range of the derivative of a differentiable bump},
journal = {Comptes Rendus. Math\'ematique},
pages = {189--194},
publisher = {Elsevier},
volume = {334},
number = {3},
year = {2002},
doi = {10.1016/S1631-073X(02)02223-9},
language = {en},
url = {http://www.numdam.org/articles/10.1016/S1631-073X(02)02223-9/}
}
TY - JOUR AU - Gaspari, Thierry TI - The range of the derivative of a differentiable bump JO - Comptes Rendus. Mathématique PY - 2002 SP - 189 EP - 194 VL - 334 IS - 3 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/S1631-073X(02)02223-9/ DO - 10.1016/S1631-073X(02)02223-9 LA - en ID - CRMATH_2002__334_3_189_0 ER -
%0 Journal Article %A Gaspari, Thierry %T The range of the derivative of a differentiable bump %J Comptes Rendus. Mathématique %D 2002 %P 189-194 %V 334 %N 3 %I Elsevier %U http://www.numdam.org/articles/10.1016/S1631-073X(02)02223-9/ %R 10.1016/S1631-073X(02)02223-9 %G en %F CRMATH_2002__334_3_189_0
Gaspari, Thierry. The range of the derivative of a differentiable bump. Comptes Rendus. Mathématique, Tome 334 (2002) no. 3, pp. 189-194. doi : 10.1016/S1631-073X(02)02223-9. http://www.numdam.org/articles/10.1016/S1631-073X(02)02223-9/
[1] Jame's theorem fails for starlike bodies, J. Functional Anal., Volume 180 (2001) no. 2, pp. 328-346
[2] The range of the gradient of a continuously differentiable bump, J. Nonlinear Convex Anal., Volume 2 (2001), pp. 1-19
[3] J.M. Borwein, M. Fabian, P.D. Loewen, The range of the gradient of a Lipschitz C1-smooth bump in infinite dimensions, Preprint, 2001
[4] Smoothness and Renormings in Banach Spaces, Pitman Monographs Surveys Pure Appl. Math., 64, 1993
[5] The Darboux property for gradients, Real Anal. Exchange, Volume 22 (1996/1997), pp. 167-173
[6] J. Saint-Raymond, Local inversion for differentiable functions and Darboux property, Preprint, 2001
[7] Introduction to Topology and Modern Analysis, Internat. Ser. Pure Appl. Math., 1963
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