Normal structure in James spaces. (English) Zbl 0883.46014

Khamsi, M. A.; Swaminathan, S.

J. Math. Anal. Appl. 204, No. 3, 765-773 (1996).

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

Keywords:

James space; normal structure; Banach space with a basis; symmetric; boundedly complete; 1-unconditional; uniformly monotone

Citations:

Zbl 0042.36102