## Normal structure in James spaces.(English)Zbl 0883.46014

Let $$X$$ be a Banach space with a basis $$(x_n)$$. Let $$c_0$$ be the space of all real sequences that converges to $$0$$. The famous James space $$J(X)$$ [R. C. James, Proc. Natl. Acad. Sci. USA 37, 174-177 (1951; Zbl 0042.36102)] consists of all sequences $$(\alpha_n)\in c_0$$ for which $\sup\Biggl\{\Biggl|\sum_{1\leq i\leq n}(\alpha_{p_i}- \alpha_{p_{i+1}}) x_i+(\alpha_{p_{n+1}}- \alpha_{p_1}) x_{n+1}\Biggr|\Biggr\}<\infty,$ the supremum being taken over all finite increasing sequences of positive numbers $$p_1,p_2,\dots, p_{n+1}$$.
In this interesting paper, the authors have proved the following important result about the normal structure property of James space $$J(X)$$.
Main Theorem: Let $$X$$ be a Banach space with a basis which is symmetric, boundedly complete, 1-unconditional, and uniformly monotone. Then $$J(X)$$ has the normal structure property.
Reviewer: Ismat Beg (Kuwait)

### MSC:

 46B20 Geometry and structure of normed linear spaces 46B25 Classical Banach spaces in the general theory 46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces

Zbl 0042.36102
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