Directions of uniform rotundity in direct sums of normed spaces |
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Authors: | M Fernández I Palacios |
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Institution: | Depto. Matem. Univ. de Extremadura, E-06071 Badajoz, Spain, ES
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Abstract: | Let {(Xi,|| · || i)}i ? I,\{(X_i,\left \| {\cdot } \right \| _{i})\}_{i\in I}, be an arbitrary family of normed spaces and let (E,|| · || E)(E,\left \| {\cdot } \right \| _{E}) be a monotonic normed space of real functions on the set I that is an ideal in \Bbb RI{\Bbb R}^I. We prove a sufficient condition for the direct sum space E(Xi) to be uniformly rotund in a direction. We show that this condition is also necessary for E=l¥E=\ell _{\infty }, and it is not necessary for E=l1E=\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(Xi)E(X_i) is uniformly rotund in every direction if and only if so are all the Xi. |
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