Duality and o − O-structure in non reflexive Banach spaces

Abstract

Let E be a Banach space with a supremum type norm induced by a collection of functionals  ℒ ⊂ X^* where X is a reflexive Banach space. Familiar spaces of this type are BMO, BV, C ^0,α(0<α≤1), L^q,∾, for q>1. For most of these spaces E, the predual E_* exists and can be defined by atomic decomposition of its elements. Another typical result, when it is possible to define a rich vanishing subspace E₀ ⊂ E is the "two star theorem", namely (E₀)*=E_*. This fails for E=BV and E=C^0,1=Lip.