On the SWCG property in Lebesgue-Bochner spaces
A Banach space \(X\) is called strongly weakly compactly generated (SWCG) if there is a weakly compact subset \(K_0\subseteq X\) such that every weakly compact set \(K\subseteq X\) is generated by \(K_0\) in the sense that for each \(\varepsilon>0\) there is an \(n\in \mathbb{N}\) with \(K\subseteq nK_0+\varepsilon B_X\) (where \(B_X\) denotes the closed unit ball of \(X\)). The author studies this property (which was introduced by \textit{G. Schlüchtermann} and \textit{R. F. Wheeler} [Math. Z. 199, No. 3, 387--398 (1988; Zbl 0637.46011)]) in Lebesgue-Bochner spaces \(L^p(\mu,X)\) for a probability measure \(\mu\). Among other things, it is proved that \(L^1(\mu,X)\) is an SWCG space if and only if there is a uniformly bounded, weakly compact subset \(C_0\subseteq L^1(\mu,X)\) which generates every uniformly bounded, weakly compact subset \(C\subseteq L^1(\mu,X)\) in the above sense, where \(C\) is called uniformly bounded if there is some \(r>0\) with \(C\subseteq rB_{L^{\infty}(\mu)}\). It is also proved that \(L^1(\mu,X)\) is SWCG if \(X\) is an SWCG subspace of \(L^1(\nu)\) for some measure \(\nu\). The question whether \(L^1(\mu,X)\) is SWCG for every SWCG space \(X\) is still open. In the case \(1<p<\infty\) it is proved that \(L^p(\mu,X)\) is SWCG if and only if \(X\) is reflexive (provided that \(\mu\) is non-trivial).
strongly weakly compactly generated spaces
scientific article; zbMATH DE number 6509444
scientific article