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Bump Functions with Hölder Derivatives |
Canad. J. Math. 56(2004), 699-715
Published:2004-08-01
Printed: Aug 2004
Printed: Aug 2004
- Thierry Gaspari
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 |
| MSC Classifications: |
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 |