Bases of type 𝑃 and reflexivity of Banach spaces

Holub, J. R.

doi:10.1090/S0002-9939-1970-0259567-X

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
DOI: https://doi.org/10.1090/S0002-9939-1970-0259567-X
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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