An approach to localized weak precompactness in Banach spaces (Q2778287)

The author studies localized weak precompactness in Banach spaces without invoking such deep results as Rosenthal's characterization of weak precompactness (Rosenthal's \(\ell^1\)-theorem) and Fremlin's result for perfect measure spaces (Fremlin's subsequences theorem). Let \(A\) be a bounded subset of a Banach space \(X\) and let \(K\) be a weak\(^*\)-compact subset of \(X^*\). The main theorem gives a number of equivalent statements for the fact that \(A\) is \(K\)-weakly precompact.