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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Two-norm spaces and decompositions of Banach spaces. II

Authors: P. K. Subramanian and S. Rothman
Journal: Trans. Amer. Math. Soc. 181 (1973), 313-327
MSC: Primary 46B15
MathSciNet review: 0320719
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Abstract: 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.

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Keywords: Schauder basis, Schauder decomposition, reflexive, pseudo reflexive, two-norm space, spaces with mixed topology
Article copyright: © Copyright 1973 American Mathematical Society