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

Casazza, P.

doi:10.1090/S0002-9939-1972-0315409-7

On a geometric condition related to boundedly complete bases and normal structure in Banach spaces
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Proc. Amer. Math. Soc. 36 (1972), 443-447 Request permission

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 $\left \| {\sum \nolimits _{i = 1}^n {{\alpha _i}{x_i}} } \right \| = 1$ and $\left \| {\sum \nolimits _{i = n + 1}^\infty {{\alpha _i}{x_i}} } \right \| \geqq C$ imply $\left \| {\sum \nolimits _{i = 1}^\infty {{\alpha _i}{x_i}} } \right \| \geqq 1 + {r_c}$. 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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