Weak normal structure in Banach spaces with symmetric norm
Let \(X\) be a Banach space with weak normal structure. Then, \(X\) has the fixed-point property; that is, every non-expansive map from a weakly compact subset into itself has a fixed point, which was proved by \textit{W. A. Kirk} [Am. Math. Monthly 72, 1004-1006 (1995; Zbl 0141.32402)]. In this note, the authors claim the following theorem: Let \(X\) be a separable Banach space. Then \(X\) can be renormed to fail to have weak normal structure if and only if \(X\) is not a Schur space. A Schur space is a Banach space in which a sequence converges weakly if and only if it converges in norm.
scientific article; zbMATH DE number 1340099