A Neighborly Property for the Canonical Decomposition of Transfinite Duals

For every nonreflexive Banach space X, every sequence (x*n*)n ≥ 0 ⊂ X**, and every integer k ≥ 1, we have the following inequality: [formula] (i.e., the norm decreases if x*k* is replaced by x**k± 1), where S = i(ω)X is the transfinite transpose of the canonical embedding iX of X into X**, and Sn is the nth power of S. It turns out that these inequalities are particular cases of more general ones, and that they are "optimal"-in a sense that will be made precise-if and only if X**/X is reflexive. These inequalities are shown to be very much connected to both Bellenot′s and Perrot′s work on equal sign additivity, on one side, and to the author′s work on James-type spaces, on the other side. We also apply these inequalities to study some topological and geometrical properties of a class of Banach spaces that are isometric to their biduals.