Conditional weak compactness and weak sequential completeness in vector-valued function spaces

Let E be an ideal of L-0 over a finite measure space (Omega, Sigma, mu) and let (X, parallel to . parallel to(X)) be a real Banach space. Let E(X) be the subspace of L-0(X) of mu-equivalence classes of all strongly Sigma-measurable functions f :Omega -> X consisting of all those f is an element of L-0(X) for which the scalar function parallel to f(.)parallel to(x) belongs to E. Let E(X)(n)(similar to) stand for the order continuous dual of E(X), i.e., E(X)(n)(similar to) consists of all linear functionals F on E(X) such that for a net (f(alpha)) in E(X), parallel to f(alpha)(.)parallel to(X) ->((o)) 0 in E implies F(f(alpha)) -> 0. We derive several results concerning conditional sigma(E(X)(n)(similar to), E(X))-compactness in E(X)(n)(similar to). It is shown that the space L-infinity(X)(n)(similar to) is sigma(L-infinity(X)(n)(similar to), L-infinity(X))-sequentially complete. We obtain a characterization of relatively sigma(L-infinity(X)(n)(similar to), L-infinity(X))-sequentially compact sets in (L-infinity(X)(n)(similar to). (C) 2011 Royal Netherlands Academy of Arts and Sciences. Published by Elsevier BAT. All rights reserved.