We show that, if μ is a probability measure and X is a Banach space, then the space L¹(μ,X) of Bochner integrable functions admits an equivalent Gâteaux (or uniformly Gâteaux) smooth norm provided that X has such a norm, and that if X admits an equivalent Fréchet (resp. uniformly Fréchet) smooth norm, then L¹(μ,X) has an equivalent renorming whose restriction to every reflexive subspace is Fréchet (resp. uniformly Fréchet) smooth.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm209-3-4,
author = {Mari\'an Fabian and Sebasti\'an Lajara},
title = {Smooth renormings of the Lebesgue-Bochner function space L$^1$($\mu$,X)},
journal = {Studia Mathematica},
volume = {209},
year = {2012},
pages = {247-265},
zbl = {1262.46007},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm209-3-4}
}
Marián Fabian; Sebastián Lajara. Smooth renormings of the Lebesgue-Bochner function space L¹(μ,X). Studia Mathematica, Tome 209 (2012) pp. 247-265. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm209-3-4/