ON GEOMETRIC AND TOPOLOGICAL PROPERTIES OF THE CLASSES OF HEREDITARILY $\ell_{p}$ BANACH SPACES

Abstract

A class of hereditarily $\ell_{p}$ ($1 \leq p \lt \infty$) Banach sequence spaces is constructed and denoted by $X_{\alpha,p}$. Any constructed space is a dual space. We show that (i) the predual of any member $X$ of the class of $X_{\alpha,1}$ contains asymptotically isometric copies of $c_0$.(ii) Every infinite dimensional subspace of $X$ contains asymptotically isometric complemented copies of $\ell_{1}$, and consequently, the dual X$^*$ of X contains subspaces isometrically isomorphic to $C[0,1]^*$. (iii) Every member of the class of $X_{\alpha,p}$ ($1 \leq p \lt \infty$) fails the Dunford-Pettis property. (iv) We observe that all $X_{\alpha,p}$ spaces are Banach spaces without unconditional basis but all constructed spaces contain a subspace which is weakly sequentially complete with an unconditional basis which is weakly null sequence but not in norm. (v) All spaces have asymptotic-norming and Kadec-Klee property. The predual of any $X_{\alpha,p}$ is an Asplund space.