# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## Smooth renormings of the Lebesgue–Bochner function space $L^1(\mu ,X)$

### 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.

#### Autorzy

• Marián FabianInstitute of Mathematics
Žitná 25
115 67 Praha 1, Czech Republic
e-mail