Smooth renormings of the Lebesgue–Bochner function space
Tom 209 / 2012
Studia Mathematica 209 (2012), 247-265
MSC: 46B03, 46B20, 46E40.
DOI: 10.4064/sm209-3-4
Streszczenie
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.
