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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On a geometric condition related to boundedly complete bases and normal structure in Banach spaces


Author: P. Casazza
Journal: Proc. Amer. Math. Soc. 36 (1972), 443-447
MSC: Primary 46B15
MathSciNet review: 0315409
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Abstract: A basis $ \{ {x_n}\} $ of a Banach space X is said to satisfy property A if for every number $ C > 0$ there exists a number $ {r_c} > 0$ such that $ \left\Vert {\sum\nolimits_{i = 1}^n {{\alpha _i}{x_i}} } \right\Vert = 1$ and $ \left\Vert {\sum\nolimits_{i = n + 1}^\infty {{\alpha _i}{x_i}} } \right\Vert \geqq C$ imply $ \left\Vert {\sum\nolimits_{i = 1}^\infty {{\alpha _i}{x_i}} } \right\Vert \geqq 1 + {r_c}$. It is known that property A implies: (1) $ \{ {x_n}\} $ is a boundedly complete basis of X, and (2) every convex, weakly compact subset of X has normal structure. In this paper, we construct a reflexive Banach space X, with an unconditional basis $ \{ {x_n}\} $, such that: (a) X has normal structure, and (b) there does not exist an equivalent norm on X with respect to which $ \{ {x_n}\} $ satisfies property A; showing that the converse of (1) and (2) is invalid even with the weaker conclusion that $ \{ {x_n}\} $ be equivalent to a basis satisfying property A.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1972-0315409-7
PII: S 0002-9939(1972)0315409-7
Keywords: Schauder basis, uniformly convex, locally uniformly convex, normal structure, fixed point
Article copyright: © Copyright 1972 American Mathematical Society