The weak topology on Lp of a vector measure

AbstractLet ν be a countably additive measure defined on a measurable space (Ω,Σ) and taking values in a Banach space X. Let 1<p<∞. In this paper we study some aspects of the weak topology on the Banach lattice Lp(ν) of all (equivalence classes of) measurable real-valued functions on Ω which are pth power integrable with respect to ν. We show that every subspace of Lp(ν) is weakly compactly generated and has weakly compactly generated dual. We prove that a bounded net (fα) in Lp(ν) is weakly convergent to f∈Lp(ν) if and only if ∫Afαdν→∫Afdν weakly in X for every A∈Σ. Finally, we also provide sufficient conditions ensuring that the set of functionals{f↦∫Ωfgd〈x∗,ν〉:g∈BLq(ν),x∗∈BX∗}⊂BLp(ν)∗ is a James boundary, where 1/p+1/q=1