Characterizations of almost transitive superreflexive Banach spaces

Julio Becerra Guerrero; Angel Rodriguez Palacios

Commentationes Mathematicae Universitatis Carolinae (2001)

  • Volume: 42, Issue: 4, page 629-636
  • ISSN: 0010-2628

Abstract

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Almost transitive superreflexive Banach spaces have been considered in [7] (see also [4] and [6]), where it is shown that such spaces are uniformly convex and uniformly smooth. We prove that convex transitive Banach spaces are either almost transitive and superreflexive (hence uniformly smooth) or extremely rough. The extreme roughness of a Banach space X means that, for every element u in the unit sphere of X , we have lim sup ∥ h ∥ → 0 ∥ u + h ∥ + ∥ u - h ∥ - 2 ∥ h ∥ = 2 . We note that, in general, the property of convex transitivity for a Banach space is weaker than that of almost transitivity.

How to cite

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Guerrero, Julio Becerra, and Palacios, Angel Rodriguez. "Characterizations of almost transitive superreflexive Banach spaces." Commentationes Mathematicae Universitatis Carolinae 42.4 (2001): 629-636. <http://eudml.org/doc/248760>.

@article{Guerrero2001,
abstract = {Almost transitive superreflexive Banach spaces have been considered in [7] (see also [4] and [6]), where it is shown that such spaces are uniformly convex and uniformly smooth. We prove that convex transitive Banach spaces are either almost transitive and superreflexive (hence uniformly smooth) or extremely rough. The extreme roughness of a Banach space $X$ means that, for every element $u$ in the unit sphere of $X$, we have \[ \limsup \_\{\Vert h\Vert \rightarrow 0\} \frac\{\Vert u+h\Vert +\Vert u-h\Vert -2\}\{\Vert h\Vert \}=2. \] We note that, in general, the property of convex transitivity for a Banach space is weaker than that of almost transitivity.},
author = {Guerrero, Julio Becerra, Palacios, Angel Rodriguez},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {convex transitive; almost transitive; superreflexive; uniformly smooth; rough norm; uniformly convex space; uniformly smooth space; rough space; surjective isometry; almost transitive Banach space},
language = {eng},
number = {4},
pages = {629-636},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Characterizations of almost transitive superreflexive Banach spaces},
url = {http://eudml.org/doc/248760},
volume = {42},
year = {2001},
}

TY - JOUR
AU - Guerrero, Julio Becerra
AU - Palacios, Angel Rodriguez
TI - Characterizations of almost transitive superreflexive Banach spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2001
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 42
IS - 4
SP - 629
EP - 636
AB - Almost transitive superreflexive Banach spaces have been considered in [7] (see also [4] and [6]), where it is shown that such spaces are uniformly convex and uniformly smooth. We prove that convex transitive Banach spaces are either almost transitive and superreflexive (hence uniformly smooth) or extremely rough. The extreme roughness of a Banach space $X$ means that, for every element $u$ in the unit sphere of $X$, we have \[ \limsup _{\Vert h\Vert \rightarrow 0} \frac{\Vert u+h\Vert +\Vert u-h\Vert -2}{\Vert h\Vert }=2. \] We note that, in general, the property of convex transitivity for a Banach space is weaker than that of almost transitivity.
LA - eng
KW - convex transitive; almost transitive; superreflexive; uniformly smooth; rough norm; uniformly convex space; uniformly smooth space; rough space; surjective isometry; almost transitive Banach space
UR - http://eudml.org/doc/248760
ER -

References

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  8. Franchetti C., Paya R., Banach spaces with strongly subdifferentiable norm, Bolletino U.M.I. 7-B (1993), 45-70. (1993) Zbl0779.46021MR1216708
  9. Giles J.R., Gregory D.A., Sims B., Characterisation of normed linear spaces with Mazur's intersection property, Bull. Austral. Math. Soc. 18 (1978), 105-123. (1978) Zbl0373.46028MR0493266
  10. Kalton N.J., Wood G.V., Orthonormal systems in Banach spaces and their applications, Math. Proc. Cambridge Philos. Soc. 79 (1976), 493-510. (1976) Zbl0327.46022MR0402471
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