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Bibasic sequences and norming basic sequences


Authors: William J. Davis, David W. Dean and Bor Luh Lin
Journal: Trans. Amer. Math. Soc. 176 (1973), 89-102
MSC: Primary 46B15
DOI: https://doi.org/10.1090/S0002-9947-1973-0313763-9
MathSciNet review: 0313763
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Abstract: It is shown that every infinite dimensional Banach space X contains a basic sequence $ ({x_n})$ having biorthogonal functionals $ ({f_n}) \subset {X^\ast}$ such that $ ({f_n})$ is also basic. If $ [{f_n}]$ norms $ [{x_n}]$ then $ ({f_n})$ is necessarily basic. If $ [{f_n}]$ norms $ [{x_n}]$ then $ [{x_n}]$ norms $ [{f_n}]$. In order that $ [{f_n}]$ norms $ [{x_n}]$ it is necessary and sufficient that the operators $ {S_n}x = \Sigma _1^n{f_i}(x){x_i}$ be uniformly bounded. If $ [{f_n}]$ norms $ [{x_n}]$ then $ {X^\ast}$ has a complemented subspace isomorphic to $ {[{x_n}]^\ast}$. Examples are given to show that $ ({f_n})$ need not be basic and, if $ ({f_n})$ is basic, still $ [{f_n}]$ need not norm $ [{x_n}]$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1973-0313763-9
Keywords: Basis, Markushevich basis, reflexivity, projection, complemented subspace, pseudo-reflexive space
Article copyright: © Copyright 1973 American Mathematical Society

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