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Smooth Partitions of Unity on Certain C(K) Spaces

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  21 December 2009

Petr Hájek
Petr Hájek
Affiliation:
Mathematical Institute, Czech Academy of Science, Žitná 25, Praha, 11567, Czech Rep. E-mail: [email protected]
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Extract

One of the main open problems in the theory of Asplund spaces is whether every Asplund space admits a Fréchet differentiable bump function. This problem is also open for C(K) Asplund spaces, where it is unknown even for C∞-Fréchet smooth bump (a general Asplund space does not always admit C2-Fréchet smooth bump – it suffices to consider ℓ3/2[DGZ2]).

MSC classification

Secondary: 46B03: Isomorphic theory (including renorming) of Banach spaces 46B10: Duality and reflexivity
Type
Research Article
Information
Mathematika , Volume 52 , Issue 1-2 , December 2005 , pp. 131 - 138
Copyright
Copyright © University College London 2005

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References

[D]Deville, R., Problemes de renormages, J. Funct. Anal. 68 (1986), 117–129.Google Scholar
[DGZ1]Deville, R., Godefroy, G. and Zizler, V., The three space problem for smooth partitions of unity and C(K) spaces, Math. Anal. 288 (1990), 613–625.CrossRefGoogle Scholar
[DGZ2]Deville, R., Godefroy, G. and Zizler, V., Smoothness and renormings in Banach spaces Longman (1993).Google Scholar
[DPWZ]Godefroy, G., Pelant, J., Whitefield, J.H.M. and Zizler, V., Banach space properties of Ciesielski-Pol's C(K), Proc. Amer. Math. Soc. 103 (1988), 1087–1094.Google Scholar
[H1]Haydon, R., Smooth functions and partitions of unity on certain Banach spaces, Quart. J. Math. Oxford 47 (1996), 455–468.Google Scholar
[H2]Haydon, R., Trees in renorming theory, Proc. London Math. Soc. 78 (1999), 549–584.CrossRefGoogle Scholar
[T]Torunczyk, H., Smooth partitions of unity on some nonseparable spaces, Studia Math. 46 (1973), 43–51.CrossRefGoogle Scholar