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Bump Functions with Hölder Derivatives

  Published:2004-08-01
 Printed: Aug 2004
  • Thierry Gaspari
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Abstract

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 derivative Banach space, bump function, range of the derivative
MSC Classifications: 46T20, 26E15, 26B05 show english descriptions Continuous and differentiable maps [See also 46G05]
Calculus of functions on infinite-dimensional spaces [See also 46G05, 58Cxx]
Continuity and differentiation questions
46T20 - Continuous and differentiable maps [See also 46G05]
26E15 - Calculus of functions on infinite-dimensional spaces [See also 46G05, 58Cxx]
26B05 - Continuity and differentiation questions
 

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