Conditional weak compactness and weak sequential completeness in vector-valued function spaces

Let EE be an ideal of L0L0 over a finite measure space (Ω,Σ,μ)(Ω,Σ,μ) and let (X,‖⋅‖X)(X,‖⋅‖X) be a real Banach space. Let E(X)E(X) be the subspace of L0(X)L0(X) of μμ-equivalence classes of all strongly ΣΣ-measurable functions f:Ω→Xf:Ω→X consisting of all those f∈L0(X)f∈L0(X) for which the scalar function ‖f(⋅)‖X‖f(⋅)‖X belongs to EE. Let E(X)n∼ stand for the order continuous dual of E(X)E(X), i.e., E(X)n∼ consists of all linear functionals FF on E(X)E(X) such that for a net (fα)(fα) in E(X)E(X), ‖fα(⋅)‖X⟶(o)0 in EE implies F(fα)⟶0F(fα)⟶0. We derive several results concerning conditional σ(E(X)n∼,E(X))-compactness in E(X)n∼. It is shown that the space L∞(X)n∼ is σ(L∞(X)n∼,L∞(X))-sequentially complete. We obtain a characterization of relatively σ(L∞(X)n∼,L∞(X))-sequentially compact sets in L∞(X)n∼.