Rotundity in Lebesgue-Bochner function spaces

Smith, Mark A.; Turett, Barry

doi:10.1090/S0002-9947-1980-0549157-4

Rotundity in Lebesgue-Bochner function spaces
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by Mark A. Smith and Barry Turett

Trans. Amer. Math. Soc. 257 (1980), 105-118

DOI: https://doi.org/10.1090/S0002-9947-1980-0549157-4

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Abstract:

This paper concerns the isometric theory of the Lebesgue-Bochner function space ${L^p}(\mu , X)$ where $1 < p < \infty$. Specifically, the question of whether a geometrical property lifts from X to ${L^p} (\mu , X)$ 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 ${L^p} (\mu , X)$. Consequently, Lebesgue-Bochner function spaces with the Radon-Riesz property are examined more closely.

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