# On Gateaux differentiable bump functions

Francisco Hernández; Stanimir Troyanski

Studia Mathematica (1996)

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

## Access Full Article

top## Abstract

top## How to cite

topHernández, 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

top- [BF] N. Bonic and J. Frampton, Smooth functions on Banach manifolds, J. Math. Mech. 15 (1966), 877-898. Zbl0143.35202
- [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.
- [DU] J. Diestel and J. J. Uhl, Vector Measures, Math. Surveys 15, Amer. Math. Soc., 1977.
- [E] I. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc. 1 (1979), 443-474. Zbl0441.49011
- [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.
- [FWZ] M. Fabian, J. Whitfield and V. Zizler, Norms with locally Lipschitzian derivatives, Israel J. Math. 44 (1983), 262-276. Zbl0521.46009
- [GJ] R. Gonzalo and J. Jaramillo, Smoothness and estimates of sequences in Banach spaces, ibid. 89 (1995), 321-341. Zbl0823.46013
- [Ph] R. Phelps, Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Math. 1364, Springer, New York, 1989. Zbl0658.46035
- [M] R. Maleev, Higher order uniformly Gateaux differentiable norms in Orlicz spaces, Rocky Mountain J. Math. 25 (1995), 1117-1136. Zbl0843.46020
- [LT] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Springer, New York, 1979. Zbl0403.46022
- [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
- [T] S. Troyanski, Gateaux differentiable norms in ${L}_{p}$, Math. Ann. 287 (1990), 221-227. Zbl0675.46010
- [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

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.