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A class of separable Banach sequence spaces is constructed. A member X of this class (i) is a hereditarily I 1 dual space which fails the Schur property, and (ii) is of codimension one in its first Baire class. A consequence of (ii) is that X is not isomorphic to the square of any Banach space Y.Introduction. In this paper we introduce and study a new class of Banach sequence spaces, the X a spaces. The definition of the norm in a particular X a space depends on the action of special sequences of intervals of integers on a vector x = (t l9 t 2 , -) (as in the definition of the James space / [6]) in conjunction with a fixed sequence of weighting factors (as in the Lorentz sequence spaces [7].) Let X denote a specific X a space, and let (e t ) denote the sequence of usual unit vectors in X (i.e. e t (j) = 8 tJ for integers / and j). Our main result is the following:

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