Two-Norm Spaces and Decompositions of Banach Spaces. II

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