On uniformly Gâteaux smooth C ( n ) -smooth norms on separable Banach spaces

Marián J. Fabián; Václav Zizler

Czechoslovak Mathematical Journal (1999)

  • Volume: 49, Issue: 3, page 657-672
  • ISSN: 0011-4642

Abstract

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Every separable Banach space with C ( n ) -smooth norm (Lipschitz bump function) admits an equivalent norm (a Lipschitz bump function) which is both uniformly Gâteaux smooth and C ( n ) -smooth. If a Banach space admits a uniformly Gâteaux smooth bump function, then it admits an equivalent uniformly Gâteaux smooth norm.

How to cite

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Fabián, Marián J., and Zizler, Václav. "On uniformly Gâteaux smooth $C^{(n)}$-smooth norms on separable Banach spaces." Czechoslovak Mathematical Journal 49.3 (1999): 657-672. <http://eudml.org/doc/30513>.

@article{Fabián1999,
abstract = {Every separable Banach space with $C^\{(n)\}$-smooth norm (Lipschitz bump function) admits an equivalent norm (a Lipschitz bump function) which is both uniformly Gâteaux smooth and $C^\{(n)\}$-smooth. If a Banach space admits a uniformly Gâteaux smooth bump function, then it admits an equivalent uniformly Gâteaux smooth norm.},
author = {Fabián, Marián J., Zizler, Václav},
journal = {Czechoslovak Mathematical Journal},
keywords = {separable Banach space; uniformly Gâteaux smooth; -smooth; integral convolution; bump function},
language = {eng},
number = {3},
pages = {657-672},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On uniformly Gâteaux smooth $C^\{(n)\}$-smooth norms on separable Banach spaces},
url = {http://eudml.org/doc/30513},
volume = {49},
year = {1999},
}

TY - JOUR
AU - Fabián, Marián J.
AU - Zizler, Václav
TI - On uniformly Gâteaux smooth $C^{(n)}$-smooth norms on separable Banach spaces
JO - Czechoslovak Mathematical Journal
PY - 1999
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 49
IS - 3
SP - 657
EP - 672
AB - Every separable Banach space with $C^{(n)}$-smooth norm (Lipschitz bump function) admits an equivalent norm (a Lipschitz bump function) which is both uniformly Gâteaux smooth and $C^{(n)}$-smooth. If a Banach space admits a uniformly Gâteaux smooth bump function, then it admits an equivalent uniformly Gâteaux smooth norm.
LA - eng
KW - separable Banach space; uniformly Gâteaux smooth; -smooth; integral convolution; bump function
UR - http://eudml.org/doc/30513
ER -

References

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  2. Calcul différentiel formes différentielles, Herman, Paris 1967. MR0223194
  3. Smoothness and renormings in Banach spaces, Pitman Monographs, No. 64, Longman House, Harlow, 1993. (1993) MR1211634
  4. 10.1007/BF02760975, Israel J. Math. 44 (1983), 262–276. (1983) MR0693663DOI10.1007/BF02760975
  5. 10.1017/S000497270001412X, Bull. Australian Math. Soc. 51 (1995), 291–300. (1995) MR1322795DOI10.1017/S000497270001412X
  6. Convex functions, monotone operators and differentiability, Lecture Notes in Math. No. 1364, Springer Verlag, 1993. (1993) Zbl0921.46039MR1238715
  7. Uniformly differentiable bump functions, Preprint. Zbl0876.46007MR1421846

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