On weighted James' spaces (Q1915572)

The topological structure of weighted James spaces \(J(h)\) is considered in this note. The main result states that \(J(h)\) is isomorphic to the classical James space \(J\) if and only if the weight \(h\) is bounded, in other case \(J(h)\) is reflexive. For a non-decreasing weight \(h\) it is proved that \(J(h)\) is isomorphic to one of the following three spaces: (i) \(J\); (ii) \(\ell_2(J_1, J_2,\dots)\) (where \(J_n\) is the subspace of \(J\) spanned by the first \(n\) basis vectors); (iii) \(\ell_2\).