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.
@article{bwmeta1.element.bwnjournal-article-smv101i1p69bwm,
author = {Mari\'an Fabian},
title = {On a dual locally uniformly rotund norm on a dual Va\v s\'ak space},
journal = {Studia Mathematica},
volume = {100},
year = {1991},
pages = {69-81},
zbl = {0815.46018},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv101i1p69bwm}
}
Fabian, Marián. On a dual locally uniformly rotund norm on a dual Vašák space. Studia Mathematica, Tome 100 (1991) pp. 69-81. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv101i1p69bwm/
[00000] [1] D. Amir and J. Lindenstrauss, The structure of weakly compact sets in Banach spaces, Ann. of Math. 88 (1968), 35-46. | Zbl 0164.14903
[00001] [2] J. Diestel, Geometry of Banach Spaces, Selected Topics, Lecture Notes in Math. 485, Springer, Berlin 1975. | Zbl 0307.46009
[00002] [3] N. Dunford and J. T. Schwartz, Linear Operators I, Interscience Publ., New York 1958. | Zbl 0084.10402
[00003] [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. | Zbl 0764.46012
[00004] [5] G. Godefroy, Existence de normes très lisses sur certains espaces de Banach, Bull. Sci. Math. (2) 106 (1982), 63-68. | Zbl 0514.46012
[00005] [6] G. Godefroy, S. Troyanski, J. Whitfield, and V. Zizler, Smoothness in weakly compactly generated Banach spaces, J. Funct. Anal. 52 (1983), 344-352. | Zbl 0517.46010
[00006] [7] S. Mercourakis, On weakly countably determined Banach spaces, Trans. Amer. Math. Soc. 300 (1987), 307-327. | Zbl 0621.46018
[00007] [8] S. Mercourakis, A dual weakly K-analytic Banach space is not necessarily a subspace of a weakly compactly generated Banach space, a manuscript.
[00008] [9] M. Talagrand, Espaces de Banach faiblement K-analytiques, Ann. of Math. 110 (1979), 407-438. | Zbl 0393.46019
[00009] [10] S. Troyanski, On locally uniformly convex and differentiable norms in certain nonseparable Banach spaces, Studia Math. 37 (1971), 173-180. | Zbl 0214.12701
[00010] [11] L. Vašák, On one generalization of weakly compactly generated Banach spaces, ibid. 70 (1981), 11-19. | Zbl 0376.46012