On the space Lp(μ,X). (Spanish: Sobre el espacio Lp(μ,X)).

Abstract

Letting X be a Banach space, (S,μ,Σ) a space of finite measure, 1<p<∞ , and L p (μ,X) the space of functions from S into X with |f| p integrable, the author studies the dual pair (L p (μ,X),L q (μ,X ′ )). He gives necessary conditions, which are also sufficient if X has the Radon-Nikodým property, in order that a subset of L p (μ,X) be σ(L p (μ,X),L q (μ,X ′ )) -relatively sequentially compact and shows that L p (μ,X) is σ(L p (μ,X),L q (μ,X ′ )) sequentially complete for every finite measure μ if and only if X is weakly sequentially complete and has the Radon-Nikodým property.