Geometry of Lebesgue-Bochner function spaces—smoothness

Leonard, I. E.; Sundaresan, K.

doi:10.1090/S0002-9947-1974-0367652-5

Geometry of Lebesgue-Bochner function spaces—smoothness
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by I. E. Leonard and K. Sundaresan PDF

Trans. Amer. Math. Soc. 198 (1974), 229-251 Request permission

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.

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