On the nonseparable subspaces of \(J(\eta)\) and \(C([1,\eta])\) (Q2706838)

The authors answer natural questions about the structure of the Banach spaces \(J(\eta)\) and \(C([1,\eta])\). For example in 1967, \textit{R. Herman} and \textit{H. Whitley} [Stud. Math. 28, 289-294 (1967; Zbl 0148.37101)] proved that every closed, infinite dimensional subspace of the James space, \(J\), contains a copy of \(\ell_2\). In 1977 \textit{P. G. Casazza, B. Lin} and \textit{R. H. Lohman} [Proc. Am. Math. Soc. 67 (1977), 265-271 (1978; Zbl 0377.46008)] proved that every closed, infinite dimensional subspace of \(J\) contains a complemented copy of \(\ell_2\). Then in 1978 \textit{J. Hagler} and \textit{E. Odell} [Ill. J. Math 22, 290-294 (1978; Zbl 0391.46015)] proved that if \(\eta\) is a natural cardinal then every closed, infinite dimensional subspace of the Long James Space, \(J(\eta)\) contains a copy of \(\ell_2\). Here the authors complete the progression by showing that every closed, infinite dimensional subspace of \(J(\eta)\), that has density character \(\eta\), contains a complemented copy of \(\ell_2(\eta)\). NEWLINENEWLINENEWLINE A similar result proves the existence of complemented copies of \(c_0(\eta)\) in \(C([1,\eta])\). NEWLINENEWLINENEWLINEThese structure properties are applied to give examples that delineate the Banach spaces with the hereditary density property and those that are weakly Lindelöf determined.