Normal structure in James spaces
Let \(X\) be a Banach space with a basis \((x_n)\). Let \(c_0\) be the space of all real sequences that converges to \(0\). The famous James space \(J(X)\) [\textit{R. C. James}, Proc. Natl. Acad. Sci. USA 37, 174-177 (1951; Zbl 0042.36102)] consists of all sequences \((\alpha_n)\in c_0\) for which \[ \sup\Biggl\{\Biggl|\sum_{1\leq i\leq n}(\alpha_{p_i}- \alpha_{p_{i+1}}) x_i+(\alpha_{p_{n+1}}- \alpha_{p_1}) x_{n+1}\Biggr|\Biggr\}<\infty, \] the supremum being taken over all finite increasing sequences of positive numbers \(p_1,p_2,\dots, p_{n+1} \). In this interesting paper, the authors have proved the following important result about the normal structure property of James space \(J(X)\). Main Theorem: Let \(X\) be a Banach space with a basis which is symmetric, boundedly complete, 1-unconditional, and uniformly monotone. Then \(J(X)\) has the normal structure property.
scientific article; zbMATH DE number 1005598