Geometry of Lebesgue-Bochner function spaces―smoothness

Abstract

There exist real Banach spaces $E$ such that the norm in $E$ is of class ${C^\infty }$ away from zero; however, for any $p,1 \leqslant p \leqslant \infty$, the norm in the Lebesgue-Bochner function space ${L_p}(E,\mu )$ is not even twice differentiable away from zero. The main objective of this paper is to give a complete determination of the order of differentiability of the norm function in this class of Banach spaces.