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2 Bulletin of the Polish Academy of Sciences. Mathematics

2 Nowak M.

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Weak compactness in Köthe-Bochner spaces and regular methods of summability

Nowak M.

Bulletin of the Polish Academy of Sciences. Mathematics

2002

Vol. 50, no 4

417-425

Let (E, || · ||E) be a Banach function space, and let (X, || · ||X) be a Banach space. Let. E(X)ñ stand for the order continuous dual of a Köthe-Bochner space E(X) (i.e. E(X)ñ consists of all linear functionals F on E(X) such that for a net (ƒσ) in E(X), || ƒσ (·) ||x [formula] in E implies F(ƒσ) → 0). We present a characterization of conditionally σ (E(X), E(X)ñ)-compact and relatively σ (E(X), E(X)ñ) compact subsets of E(X) in terms of regular methods of summability. We generalize S. Diaz's criteria for conditional weak compactness and relative weak compactness in L1 (X).

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.