On Orlicz spaces of vector-valued functions. (Spanish: Sobre los espacios de Orlicz de funciones vectoriales)

Abstract

Let (S,Σ,μ) be a finite measure space, X a Banach space and Φ a Young function with complementary function Ψ. There is a natural duality between the Orlicz spaces LΦ(X) and LΨ(X∗), given by (f,g)↦∫⟨f,g⟩dμ. Assume that X satisfies the Radon-Nikodým property. One of the main results obtained in this paper is the following: K⊂LΦ(X) is σ(LΦ(X),LΨ(X∗)) relatively sequentially compact if and only if the following conditions are satisfied: (i) K is norm-bounded, (ii) the set K(A)={∫Afdμ:f∈K} is relatively weakly compact in X for every A∈Σ, and (iii) limμ(A)→0sup{∫A⟨f,g⟩dμ:f∈K}=0 for every g∈LΨ(X∗).