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Pyramidal vectors and smooth functions on Banach spaces


Authors: R. Deville and E. Matheron
Journal: Proc. Amer. Math. Soc. 128 (2000), 3601-3608
MSC (2000): Primary 46B20
DOI: https://doi.org/10.1090/S0002-9939-00-05519-2
Published electronically: June 7, 2000
MathSciNet review: 1694858
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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 [Enhancements On Off] (What's this?)

  • [B] S. M. Bates, On smooth, nonlinear surjections of Banach spaces, Israel J. Math. 100 (1997), 209-220. MR 98i:58016
  • [H1] P. Hájek, Smooth functions on $c_{\scriptscriptstyle 0}$, Israel J. Math. 104 (1998), 17-27. MR 99d:46063
  • [H2] P. Hájek, Smooth functions on $C(K)$, Israel J. Math. 107 (1998), 237-252. CMP 99:05

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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
Email: matheron@math.u-bordeaux.fr

DOI: https://doi.org/10.1090/S0002-9939-00-05519-2
Received by editor(s): February 19, 1999
Published electronically: June 7, 2000
Communicated by: Dale Alspach
Article copyright: © Copyright 2000 American Mathematical Society

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