Two-norm spaces and decompositions of Banach spaces. II

Abstract

Let X X be a Banach space, Y Y a closed subspace of X ∗ {X^\ast } . One says X X is Y Y -reflexive if the canonical imbedding of X X onto Y ∗ {Y^\ast } is an isometry and Y Y -pseudo reflexive if it is a linear isomorphism onto. If X X has a basis and Y Y is the closed linear span of the corresponding biorthogonal functionals, necessary and sufficient conditions for X X to be Y Y -pseudo reflexive are due to I. Singer. To every B B -space X X with a decomposition we associate a canonical two-norm space X s {X_s} and show that the properties of X s {X_s} , in particular its γ \gamma -completion, may be exploited to give different proofs of Singer’s results and, in particular, to extend them to B B -spaces with decompositions. This technique is then applied to a study of direct sum of B B -spaces with respect to a BK space. Necessary and sufficient conditions for such a space to be reflexive are obtained.