Rotundity in Lebesgue-Bochner function spaces

Abstract

This paper concerns the isometric theory of the Lebesgue-Bochner function space L p ( μ , X ) {L^p}(\mu ,\,X) where 1 > p > ∞ 1 > p > \infty . Specifically, the question of whether a geometrical property lifts from X to L p ( μ , X ) {L^p}\,(\mu ,\,X) is examined. Positive results are obtained for the properties local uniform rotundity, weak uniform rotundity, uniform rotundity in each direction, midpoint local uniform rotundity, and B-convexity. However, it is shown that the Radon-Riesz property does not lift from X to L p ( μ , X ) {L^p}\,(\mu ,\,X) . Consequently, Lebesgue-Bochner function spaces with the Radon-Riesz property are examined more closely.