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

DOI:
https://doi.org/10.1090/S0002-9947-1973-0320719-9

MathSciNet review:
0320719

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Abstract | References | Similar Articles | Additional Information

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

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1973-0320719-9

Keywords:
Schauder basis,
Schauder decomposition,
reflexive,
pseudo reflexive,
two-norm space,
spaces with mixed topology

Article copyright:
© Copyright 1973
American Mathematical Society