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2 Bulletin of the Polish Academy of Sciences. Mathematics

2 Nowak M.

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Weak sequential compactness and completeness in Koethe-Bochner spaces

Nowak M.

Bulletin of the Polish Academy of Sciences. Mathematics

1999

Vol. 47, no 3

209-220

Let (E,[...]) be a Banach function space over a probability measure space, and let (X,[...]) be a Banach space. Let E(X)[...] stand for the order continuous dual of a Koethe-Bochner space E(X ) (i.e., E(X )[...] consists of all linear functionals F on E(X ) such that for a net (f[sigma]) in E(X),[...] 0 in E implies F(f[sigma]) --> 0). We present a characterization of conditional sigma(E(X), E(X)[...])-compactness and relative sigma(E(X), E(X)[...] )-compactness in E(X ). We generalize J. Diestel, W. Ruess and W. Schachermayer's criterion for relative weak compactness in E(X ) as well as M. Talagrand's results on conditional weak compactness and weak sequential completeness in E(X ) by removing order continuity of (E, [...]). Applications to Orlicz-Bochner spaces are given.

Weak-sequential compactness on non-locally convex Orlicz spaces

Nowak M.

Bulletin of the Polish Academy of Sciences. Mathematics

1998

Vol. 46, no 3

225-231

Let [L^fi] be an Orlicz space defined by a finite valued Orlicz function [fi] (not necesarilly convex) over a [sigma]-finite atomless measure space. Let (L^fi[...] be the order continous dual of [L^fi]. It is proved that a subset Z of [L^fi] is conditionally sequentially sigma([L^fi],(L^fi])[...)-compact (i.e., every sequence in Z contains a sigma([L^[fi],(L^fi])[...])-Cauchy subsequence) if and only if Z is norm bounded in some Orlicz space [L^psi] where psi increases more rapidly than [...] (the convex minorant of fi).