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Let X denote a specific space of the class of Xα,p Banach sequence spaces which were constructed by Hagler and the first named author as classes of hereditarily p Banach spaces. We show that for p > 1 the Banach space X contains asymptotically isometric copies of p. It is known that any member of the class is a dual space. We show that the predual of X contains isometric copies of q where 1/p + 1/q = 1. For p = 1 it is known that the predual of the Banach space X contains asymptotically isometric copies of c 0 . Here we give a direct proof of the known result that X contains asymptotically isometric copies of 1 .

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