On a problem of H. P. Rosenthal (Q2716488)

H. P. Rosenthal associated with a non-reflexive separable Banach space \(X\) an ordinal index defined on the elements of the bidual, and shows that \(X\) contains no isomorph of \(c_0\) iff for any element of the bidual which is not in the space the value of the index is countable. Here, it is shown that if \(X\) contains no isomorph of \(c_0\), then in some cases the ordinal index is uniformly bounded on the elements of the bidual which are not in the space by a countable ordinal. In particular, it is true when \(X\) contains no isomorph of \(\ell_1\).