RENORMING IN THE SPACE OF BOCHNER INTEGRABLE FUNCTIONS L1(MU,X)

A sufficient condition is given when a subspace L subset-of L1(mu, X) of the space of Bochner integrable function, defined on a finite and positive measure space (S,SIGMA,mu) with values in a Banach space X, is locally uniformly convex renormable in terms of the integrable evaluation {integral A fd-mu; f is-an-element-of L}. This shows the lifting property that L1(mu, X) is renormable if and only if X is, and indicates a large class of renormable subspaces even if X does not admit an equivalent locally uniformly convex norm.