On Gateaux differentiable bump functions

Francisco Hernández; Stanimir Troyanski

Studia Mathematica (1996)

  • Volume: 118, Issue: 2, page 135-143
  • ISSN: 0039-3223

Abstract

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It is shown that the order of Gateaux smoothness of bump functions on a wide class of Banach spaces with unconditional basis is not better than that of Fréchet differentiability. It is proved as well that in the separable case this order for Banach lattices satisfying a lower p-estimate for 1≤ p < 2 can be only slightly better.

How to cite

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Hernández, Francisco, and Troyanski, Stanimir. "On Gateaux differentiable bump functions." Studia Mathematica 118.2 (1996): 135-143. <http://eudml.org/doc/216268>.

@article{Hernández1996,
abstract = {It is shown that the order of Gateaux smoothness of bump functions on a wide class of Banach spaces with unconditional basis is not better than that of Fréchet differentiability. It is proved as well that in the separable case this order for Banach lattices satisfying a lower p-estimate for 1≤ p < 2 can be only slightly better.},
author = {Hernández, Francisco, Troyanski, Stanimir},
journal = {Studia Mathematica},
keywords = {Gâteaux smoothness of bump functions; Banach spaces with unconditional basis; Fréchet differentiability; Banach lattices; lower -estimate},
language = {eng},
number = {2},
pages = {135-143},
title = {On Gateaux differentiable bump functions},
url = {http://eudml.org/doc/216268},
volume = {118},
year = {1996},
}

TY - JOUR
AU - Hernández, Francisco
AU - Troyanski, Stanimir
TI - On Gateaux differentiable bump functions
JO - Studia Mathematica
PY - 1996
VL - 118
IS - 2
SP - 135
EP - 143
AB - It is shown that the order of Gateaux smoothness of bump functions on a wide class of Banach spaces with unconditional basis is not better than that of Fréchet differentiability. It is proved as well that in the separable case this order for Banach lattices satisfying a lower p-estimate for 1≤ p < 2 can be only slightly better.
LA - eng
KW - Gâteaux smoothness of bump functions; Banach spaces with unconditional basis; Fréchet differentiability; Banach lattices; lower -estimate
UR - http://eudml.org/doc/216268
ER -

References

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  2. [DGZ] R. Deville, G. Godefroy and V. Zizler, Smoothness and Renormings in Banach Spaces, Pitman Monographs and Surveys in Pure and Appl. Math. 64, Longman Sci. & Tech., 1993. 
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  5. [FPWZ] M. Fabian, D. Preis, J. Whitfield and V. Zizler, Separating polynomials on Banach spaces, Quart. J. Math. Oxford Ser. (2) 40 (1989), 409-422. 
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  8. [Ph] R. Phelps, Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Math. 1364, Springer, New York, 1989. Zbl0658.46035
  9. [M] R. Maleev, Higher order uniformly Gateaux differentiable norms in Orlicz spaces, Rocky Mountain J. Math. 25 (1995), 1117-1136. Zbl0843.46020
  10. [LT] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Springer, New York, 1979. Zbl0403.46022
  11. [MV] D. McLaughlin and J. Vanderwerff, Higher order Gateaux smooth bump functions on Banach spaces, Bull. Austral. Math. Soc. 51 (1995), 291-300. Zbl0838.46007
  12. [T] S. Troyanski, Gateaux differentiable norms in L p , Math. Ann. 287 (1990), 221-227. Zbl0675.46010
  13. [V] J. Vanderwerff, Second order Gateaux differentiability and an isomorphic characterization of Hilbert spaces, Quart. J. Math. Oxford Ser 2, 44 (1993), 249-255. Zbl0799.46017

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