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Transactions of the American Mathematical Society

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

 

 

Geometry of Lebesgue-Bochner function spaces--smoothness


Authors: I. E. Leonard and K. Sundaresan
Journal: Trans. Amer. Math. Soc. 198 (1974), 229-251
MSC: Primary 46E40; Secondary 58C20
DOI: https://doi.org/10.1090/S0002-9947-1974-0367652-5
MathSciNet review: 0367652
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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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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0367652-5
Keywords: Lebesgue-Bochner function spaces, higher-order differentiability, smoothness
Article copyright: © Copyright 1974 American Mathematical Society