On dual of Banach sequence spaces

Abstract

J. Hagler and P. Azimi have introduced a class of Banach sequence spaces, the Xα,1 spaces as a class of hereditarily $\ell_1$ Banach spaces. In this paper, (i) We show that X*α, 1, the dual of Banach space Xα,1 contains asymptotically isometric copies of $\ell_\infty$. (ii) With two methods, we show that X*α,1 is nonseparable although Xα,1 is a separable Banach space. Also, we show Xα,1 is not hereditarily indecomposable.