
Journal of Convex Analysis 12 (2005), No. 2, 447463 Copyright Heldermann Verlag 2005 Conditional and Relative Weak Compactness in VectorValued Function Spaces Marian Nowak Faculty of Mathematics, University of Zielona Góra, ul. Szafrana 4A, 65516 Zielona Góra, Poland M.Nowak@wmie.uz.zgora.pl [Abstractpdf] \newcommand{\norm}{\vert\vert} 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 $\,E(X)$. Keywords: Vectorvalued function spaces, KoetheBochner spaces, conditional weak compactness, weak sequential compactness, weak compactness, weak sequential completeness, almost reflexivity, reflexivity, absolute weak topologies. MSC: 46E40; 46A50, 46A20, 46A25 [ Fulltextpdf (502 KB)] for subscribers only. 