Pyramidal vectors and smooth functions on Banach spaces
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- by R. Deville and E. Matheron PDF
- Proc. Amer. Math. Soc. 128 (2000), 3601-3608 Request permission
Abstract:
We prove that if $X$, $Y$ are Banach spaces such that $Y$ has nontrivial cotype and $X$ has trivial cotype, then smooth functions from $X$ into $Y$ have a kind of “harmonic" behaviour. More precisely, we show that if $\Omega$ is a bounded open subset of $X$ and $f:{\overline {\Omega }}\to Y$ is $C^{1}$-$$smooth with uniformly continuous Fréchet derivative, then $f(\partial \Omega )$ is dense in $f({\overline {\Omega }})$. We also give a short proof of a recent result of P. Hájek.References
- S. M. Bates, On smooth, nonlinear surjections of Banach spaces, Israel J. Math. 100 (1997), 209–220. MR 1469111, DOI 10.1007/BF02773641
- Petr Hájek, Smooth functions on $c_0$, Israel J. Math. 104 (1998), 17–27. MR 1622271, DOI 10.1007/BF02897057
- P. Hájek, Smooth functions on $C(K)$, Israel J. Math. 107 (1998), 237-252.
Additional Information
- R. Deville
- Affiliation: Université Bordeaux 1, 351, Cours de la libération, 33405 Talence Cedex, France
- Email: deville@math.u-bordeaux.fr
- E. Matheron
- Affiliation: Université Bordeaux 1, 351, Cours de la libération, 33405 Talence Cedex, France
- MR Author ID: 348460
- Email: matheron@math.u-bordeaux.fr
- Received by editor(s): February 19, 1999
- Published electronically: June 7, 2000
- Communicated by: Dale Alspach
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 3601-3608
- MSC (2000): Primary 46B20
- DOI: https://doi.org/10.1090/S0002-9939-00-05519-2
- MathSciNet review: 1694858