Renorming in the space of Bochner integrable functionsL 1(μ,X)

Abstract

A sufficient condition is given when a subspaceL⊂L 1(μ,X) of the space of Bochner integrable function, defined on a finite and positive measure space (S, Φ, μ) with values in a Banach spaceX, is locally uniformly convex renormable in terms of the integrable evaluations {∫ A fdμ;f∈L}. This shows the lifting property thatL 1(μ,X) is renormable if and only ifX is, and indicates a large class of renormable subspaces even ifX does not admit and equivalent locally uniformly convex norm.