On a dual locally uniformly rotund norm on a dual Vašák space

Marián Fabian

Studia Mathematica (1991)

  • Volume: 101, Issue: 1, page 69-81
  • ISSN: 0039-3223

Abstract

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We transfer a renorming method of transfer, due to G. Godefroy, from weakly compactly generated Banach spaces to Vašák, i.e., weakly K-countably determined Banach spaces. Thus we obtain a new construction of a locally uniformly rotund norm on a Vašák space. A further cultivation of this method yields the new result that every dual Vašák space admits a dual locally uniformly rotund norm.

How to cite

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Fabian, Marián. "On a dual locally uniformly rotund norm on a dual Vašák space." Studia Mathematica 101.1 (1991): 69-81. <http://eudml.org/doc/215893>.

@article{Fabian1991,
abstract = {We transfer a renorming method of transfer, due to G. Godefroy, from weakly compactly generated Banach spaces to Vašák, i.e., weakly K-countably determined Banach spaces. Thus we obtain a new construction of a locally uniformly rotund norm on a Vašák space. A further cultivation of this method yields the new result that every dual Vašák space admits a dual locally uniformly rotund norm.},
author = {Fabian, Marián},
journal = {Studia Mathematica},
keywords = {renorming method of transfer; weakly compactly generated Banach spaces; weakly -countably determined Banach spaces; locally uniformly rotund norm on a Vašák space},
language = {eng},
number = {1},
pages = {69-81},
title = {On a dual locally uniformly rotund norm on a dual Vašák space},
url = {http://eudml.org/doc/215893},
volume = {101},
year = {1991},
}

TY - JOUR
AU - Fabian, Marián
TI - On a dual locally uniformly rotund norm on a dual Vašák space
JO - Studia Mathematica
PY - 1991
VL - 101
IS - 1
SP - 69
EP - 81
AB - We transfer a renorming method of transfer, due to G. Godefroy, from weakly compactly generated Banach spaces to Vašák, i.e., weakly K-countably determined Banach spaces. Thus we obtain a new construction of a locally uniformly rotund norm on a Vašák space. A further cultivation of this method yields the new result that every dual Vašák space admits a dual locally uniformly rotund norm.
LA - eng
KW - renorming method of transfer; weakly compactly generated Banach spaces; weakly -countably determined Banach spaces; locally uniformly rotund norm on a Vašák space
UR - http://eudml.org/doc/215893
ER -

References

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  1. [1] D. Amir and J. Lindenstrauss, The structure of weakly compact sets in Banach spaces, Ann. of Math. 88 (1968), 35-46. Zbl0164.14903
  2. [2] J. Diestel, Geometry of Banach Spaces, Selected Topics, Lecture Notes in Math. 485, Springer, Berlin 1975. Zbl0307.46009
  3. [3] N. Dunford and J. T. Schwartz, Linear Operators I, Interscience Publ., New York 1958. Zbl0084.10402
  4. [4] M. Fabian and S. Troyanski, A Banach space admits a locally uniformly rotund norm if its dual is a Vašák space, Israel J. Math. 69 (1990), 214-224. Zbl0764.46012
  5. [5] G. Godefroy, Existence de normes très lisses sur certains espaces de Banach, Bull. Sci. Math. (2) 106 (1982), 63-68. Zbl0514.46012
  6. [6] G. Godefroy, S. Troyanski, J. Whitfield, and V. Zizler, Smoothness in weakly compactly generated Banach spaces, J. Funct. Anal. 52 (1983), 344-352. Zbl0517.46010
  7. [7] S. Mercourakis, On weakly countably determined Banach spaces, Trans. Amer. Math. Soc. 300 (1987), 307-327. Zbl0621.46018
  8. [8] S. Mercourakis, A dual weakly K-analytic Banach space is not necessarily a subspace of a weakly compactly generated Banach space, a manuscript. 
  9. [9] M. Talagrand, Espaces de Banach faiblement K-analytiques, Ann. of Math. 110 (1979), 407-438. Zbl0393.46019
  10. [10] S. Troyanski, On locally uniformly convex and differentiable norms in certain nonseparable Banach spaces, Studia Math. 37 (1971), 173-180. Zbl0214.12701
  11. [11] L. Vašák, On one generalization of weakly compactly generated Banach spaces, ibid. 70 (1981), 11-19. Zbl0376.46012

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