Conditional and Relative Weak Compactness in Vector-Valued Function Spaces

Let $\,E\,$ be an ideal of $\,L^{\rm o}\,$ over a $\,\sigma$-finite measure space $\,(\Omega, \Sigma, \mu)$, and let $\,(X, \norm \cdot \norm_X)\,$ be a real Banach space. Let $\,E(X)\,$ be a subspace of the space $\,L^{\rm o}(X)\,$ of $\,\mu$-equivalence classes of all strongly $\,\Sigma$-measurable functions $\,f:\; \Omega\longrightarrow X\,$ and consisting of all those $\,f\in L^{\rm o}(X)\,$ for which the scalar function $\,\norm f(\cdot) \norm_X\,$ belongs to $\,E$. Let $\,E(X)_n^{\sim}\,$ stand for the order continuous dual of $\,E(X)$. In this paper we characterize both conditionally $\,\sigma(E(X),I)$-compact and relatively $\,\sigma(E(X), I)$-sequentially compact subsets of $\,E(X)\,$ whenever $\,I\,$ is an ideal of $\,E(X)_n^{\sim}$. As an application, we obtain a characterization of almost reflexivity and reflexivity of a Banach space $\,X\,$ in terms of conditionally $\,\sigma(E(X), I)$-compact and relatively $\,\sigma(E(X), I)$-sequentially compact subsets of $\,E(X)$.