Directions of uniform rotundity in direct sums of normed spaces

Abstract

Let $\{(X_i,\left \| {\cdot } \right \| _{i})\}_{i\in I},$ be an arbitrary family of normed spaces and let $(E,\left \| {\cdot } \right \| _{E})$ be a monotonic normed space of real functions on the set I that is an ideal in ${\Bbb R}^I$ . We prove a sufficient condition for the direct sum space E(X i ) to be uniformly rotund in a direction. We show that this condition is also necessary for $E=\ell _{\infty }$ , and it is not necessary for $E=\ell _1$ . When E is either uniformly rotund in every direction and has compact order intervals, or weakly uniformly rotund respect to its evaluation functionals, we reestablish as a corollary the result that reads: $E(X_i)$ is uniformly rotund in every direction if and only if so are all the X i .