The range of the derivative of a differentiable bump
[L'image de la dérivée d'une fonction bosse différentiable]
Comptes Rendus. Mathématique, Tome 334 (2002) no. 3, pp. 189-194.

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 X * contenant 0 est l'image de la dérivée d'une bosse de classe Cp. Ensuite, les parties analytiques de X * 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  2 , f'( 2 ) 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 X * containing 0 is the range of the derivative of a Cp-bump. Next, analytic subsets of X * 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 f'( 2 ) is the closure of its interior, if f is a C2-bump on 2 .

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(02)02223-9
Gaspari, Thierry 1

1 Mathématiques pures de Bordeaux (MPB), UMR 5467 CNRS, Université Bordeaux 1, 351, cours de la Libération, 33405 Talence cedex, France
@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://archive.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://archive.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://archive.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://archive.numdam.org/articles/10.1016/S1631-073X(02)02223-9/

[1] Azagra, D.; Deville, R. Jame's theorem fails for starlike bodies, J. Functional Anal., Volume 180 (2001) no. 2, pp. 328-346

[2] Borwein, J.M.; Fabian, M.; Kortezov, I.; Loewen, P.D. 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] Deville, R.; Godefroy, G.; Zizler, V. Smoothness and Renormings in Banach Spaces, Pitman Monographs Surveys Pure Appl. Math., 64, 1993

[5] Malý, J. 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] Simmons, G.F. Introduction to Topology and Modern Analysis, Internat. Ser. Pure Appl. Math., 1963

Cité par Sources :