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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Bases of type $ P$ and reflexivity of Banach spaces


Author: J. R. Holub
Journal: Proc. Amer. Math. Soc. 25 (1970), 357-362
MSC: Primary 46.10
MathSciNet review: 0259567
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Abstract: The purpose of this paper is to give a number of characterizations of reflexive Banach spaces having Schauder bases.

We show that a Banach space $ X$ with a basis such that $ {c_0}$ is contained in either $ X$ or in $ {X^{\ast}}$ has a basis of type $ P$. It follows that a Banach space with an unconditional basis is reflexive if and only if no basis is of type $ P$. It is also shown that a Banach space with a basis is reflexive if and only if every basis converges weakly to zero.


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DOI: https://doi.org/10.1090/S0002-9939-1970-0259567-X
Keywords: Schauder basis, basic sequence, basis of type $ P$, reflexive Banach space, unconditional basis, basis of type $ w{c_0}$, weak convergence of bases
Article copyright: © Copyright 1970 American Mathematical Society