Some more uniformly convex spaces

Abstract

Let {Bi,i = l, 2, . . } be a sequence of Banach spaces, and define B = c P p \Bi} to be the space of sequences b={bi} with bi^Bi and Ml =(l>INh) 1/p < > 1<< . It is known that B, normed in this way, is also a Banach space. Boas 2 showed that if Bi = l p for all i or if Bi -L v for all i then B is uniformly convex. Since then I have shown 3 that if B{ -l vi or L pi , \<pi< oo, and if the sequence {pi} is not bounded away from 1 and oo (that is, if there do not exist Km ^M < oo with m^pi^M for all i), then B is not uniformly convex and cannot be renormed to an isomorphic uniformly convex space. One purpose of this note is to fill in the gap between these results by means of the following theorem.