Smooth renormings of the Lebesgue–Bochner function space $L^1(\mu ,X)$
Volume 209 / 2012
Studia Mathematica 209 (2012), 247-265
MSC: 46B03, 46B20, 46E40.
DOI: 10.4064/sm209-3-4
Abstract
We show that, if $\mu$ is a probability measure and $X$ is a Banach space, then the space $L^1(\mu,X)$ 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 $L^1(\mu,X)$ has an equivalent renorming whose restriction to every reflexive subspace is Fréchet (resp. uniformly Fréchet) smooth.
