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 of a Banach space *X* is said to satisfy property A if for every number there exists a number such that and imply . It is known that property A implies: (1) 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 , such that: (a) *X* has normal structure, and (b) there does not exist an equivalent norm on *X* with respect to which satisfies property A; showing that the converse of (1) and (2) is invalid even with the weaker conclusion that be equivalent to a basis satisfying property A.

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DOI:
https://doi.org/10.1090/S0002-9939-1972-0315409-7

Keywords:
Schauder basis,
uniformly convex,
locally uniformly convex,
normal structure,
fixed point

Article copyright:
© Copyright 1972
American Mathematical Society