A set-theoretic characterization of a certain class of \(l_ 2\)-saturated James type spaces (Q1911353)

For every nonempty countable subset \(L\) of the set of real numbers, a nonempty subset \(S\) of \(L\) is called a segment in \(L\) if for any \(s_1,s_2\in S\) and \(q\in L\), \(s_1\leq q\leq s_2\) implies that \(q\in S\). The James space \(J(L)\) is defined as the completion of the space of all real-valued functions with finite support in \(L\) under the norm \[ |x|=\sup \Biggl[\sum_{j=1}^k \Bigl|\sum_{q_n\in S_j}x_{qn} \Bigr|^2 \Biggr]^{1/2}\equiv\sup \Biggl[ \sum_{j=1}^k |\widehat{S}_j(x) |^2\Biggr]^{1/2} \] where the supremum is taken over all \(k\)-tuples \(\{s_j\}\) of pairwise disjoint segments in \(L\). The main result of this paper is about an \(\ell_2\)-saturation-\(c_0\)-containment dichotomy in the structure of \(J(L)\), when \(L\) is complete.