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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Rotundity in Lebesgue-Bochner function spaces

Authors: Mark A. Smith and Barry Turett
Journal: Trans. Amer. Math. Soc. 257 (1980), 105-118
MSC: Primary 46E40; Secondary 46B20
MathSciNet review: 549157
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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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Keywords: Lebesgue-Bochner function space, locally uniformly rotund, weakly uniformly rotund, B-convex, uniformly non-$ {l^1}(n)$, Radon-Riesz property
Article copyright: © Copyright 1980 American Mathematical Society