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

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(0α≤1), Lq,∾, for q1. 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=C0,1=Lip.