J. Aust. Math. Soc. 75 (2003), 413–422 ON-DIRECT SUMS OF BANACH SPACES AND CONVEXITY

Let X1; X2; : : : ; XN be Banach spaces and a continuous convex function with some appropriate conditions on a certain convex set inRN−1. Let.X1X2 XN / be the direct sum of X1; X2; : : : ; XN equipped with the norm associated with. We characterize the strict, uniform, and locally uniform convexity of.X1 X2 XN / by means of the convex function. As an application these convexities are characterized for the ‘p;q-sum.X1 X2 XN /p;q (1 < q p 1, q < 1), which includes the well-known facts for the ‘p-sum.X1 X2 XN /p in the case p D q