On the range of the derivative of a real-valued function with bounded support

T. Gaspari ¹

¹ Mathématiques Pures de Bordeaux (MPB), UMR 5467 CNRS Université Bordeaux 1 351, cours de la Libération 33400 Talence, France

Studia Mathematica, Tome 153 (2002) no. 1, pp. 81-99

Résumé

We study the set $f'(X)=\{f'(x): x \in X\}$ when $f:X\rightarrow \mathbb R $ is a differentiable bump. We first prove that for any $C^2$-smooth bump $f: \mathbb R^2 \rightarrow \mathbb R $ the range of the derivative of $f$ must be the closure of its interior. Next we show that if $X$ is an infinite-dimensional separable Banach space with a $C^p$-smooth bump $b:X\rightarrow \mathbb R $ such that $\| b ^{(p)} \| _{\infty} $ is finite, then any connected open subset of $X^{\ast}$ containing $0$ is the range of the derivative of a $C^p$-smooth bump. We also study the finite-dimensional case which is quite different. Finally, we show that in infinite-dimensional separable smooth Banach spaces, every analytic subset of $X^{\ast}$ which satisfies a natural linkage condition is the range of the derivative of a $C^1$-smooth bump. We then find an analogue of this condition in the finite-dimensional case

Détail

DOI : 10.4064/sm153-1-6

Keywords: study set rightarrow mathbb differentiable bump first prove smooth bump mathbb rightarrow mathbb range derivative closure its interior infinite dimensional separable banach space p smooth bump rightarrow mathbb infty finite connected subset ast containing range derivative p smooth bump study finite dimensional which quite different finally infinite dimensional separable smooth banach spaces every analytic subset ast which satisfies natural linkage condition range derivative smooth bump analogue condition finite dimensional case