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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).