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Search: MSC category 46T20 ( Continuous and differentiable maps [See also 46G05] )

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1. CJM 2004 (vol 56 pp. 699)

Gaspari, Thierry
Bump Functions with Hölder 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 derivative
Categories:46T20, 26E15, 26B05

2. CJM 2003 (vol 55 pp. 969)

Glöckner, Helge
Lie Groups of Measurable Mappings
We describe new construction principles for infinite-dimensional Lie groups. In particular, given any measure space $(X,\Sigma,\mu)$ and (possibly infinite-dimensional) Lie group $G$, we construct a Lie group $L^\infty (X,G)$, which is a Fr\'echet-Lie group if $G$ is so. We also show that the weak direct product $\prod^*_{i\in I} G_i$ of an arbitrary family $(G_i)_{i\in I}$ of Lie groups can be made a Lie group, modelled on the locally convex direct sum $\bigoplus_{i\in I} L(G_i)$.

Categories:22E65, 46E40, 46E30, 22E67, 46T20, 46T25

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