EBERLEIN COMPACTNESS

摘要

For finite measure space (X,A, μ), a Banach space E with E' its dual, and a relatively countably compact Q ? (L_1(E), σ(L_1(E), L_∞(E'))), entirely different proofs are given of the results that (i) Q is Eberlein compact, (ii) the closed convex hull of Q in (L_1(E), σ(L_1(E), L∞(E'))) is also compact and (iii) the closed convex hull of Q in (L_1(E), σ(L_1(E), L_∞(E'))) and in (L_1(E), ||·||1) are the same.