Eberlein compactness

Abstract

For finite measure space $(X, \A, \mu)$, a Banach space $E$ with $E^{\p}$ its dual, and a relatively countably compact $Q \subset (L_{1}(E), \sigma(L_{1}(E), L_{\infty}(E^{\p})))$, entirely different proofs are given of the results that (i)~$\overline{Q}$ is Eberlein compact, (ii)~the closed convex hull of $\overline{Q}$ in $(L_{1}(E), \sigma(L_{1}(E), L_{\infty}(E^{\p})))$ is also compact and (iii)~the closed convex hull of $\overline{Q}$ in $(L_{1}(E), \sigma(L_{1}(E), L_{\infty}(E^{\p})))$ and in $(L_{1}(E), \| \cdot\|_{1})$ are the same.