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1. CJM 2004 (vol 56 pp. 699)

Gaspari, Thierry
 Bump Functions with HÃ¶lder Derivatives We study the range of the gradients of a \$C^{1,\al}\$-smooth bump function defined on a Banach space. We find that this set must satisfy two geometrical conditions: It can not be too flat and it satisfies a strong compactness condition with respect to an appropriate distance. These notions are defined precisely below. With these results we illustrate the differences with the case of \$C^1\$-smooth bump functions. Finally, we give a sufficient condition on a subset of \$X^{\ast}\$ so that it is the set of the gradients of a \$C^{1,1}\$-smooth bump function. In particular, if \$X\$ is an infinite dimensional Banach space with a \$C^{1,1}\$-smooth bump function, then any convex open bounded subset of \$X^{\ast}\$ containing \$0\$ is the set of the gradients of a \$C^{1,1}\$-smooth bump function. Keywords:Banach space, bump function, range of the derivativeCategories:46T20, 26E15, 26B05

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