Conditional and Relative Weak Compactness in Vector-Valued Function Spaces

Let E be an ideal of Lo over a σ-finite measure space (Ω,Σ, μ), and let (X, ‖ ·‖X) be a real Banach space. Let E(X) be a subspace of the space Lo(X) of μ-equivalence classes of all strongly Σ-measurable functions f : Ω −→ X and consisting of all those f ∈ Lo(X) for which the scalar function ‖f(·)‖X belongs to E. Let E(X)n stand for the order continuous dual of E(X). In this paper we characterize both conditionally σ(E(X), I)-compact and relatively σ(E(X), I)-sequentially compact subsets of E(X) whenever I is an ideal of E(X)n . As an application, we obtain a characterization of almost reflexivity and reflexivity of a Banach space X in terms of conditionally σ(E(X), I)-compact and relatively σ(E(X), I)-sequentially compact subsets of E(X).