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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

An extremal vector-valued $ L\sp{p}$-function taking no extremal vectors as values


Author: Peter Greim
Journal: Proc. Amer. Math. Soc. 84 (1982), 65-68
MSC: Primary 46E40; Secondary 46B20
DOI: https://doi.org/10.1090/S0002-9939-1982-0633279-0
MathSciNet review: 633279
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Abstract: We give an example of a nonseparable Banach space $ V$ and a function $ x$ on [0, 1] with values in the unit sphere of $ V$ that is an extreme point of the unit balls of all Bochner $ {L^p}$-spaces $ {L^p}(\lambda ,V)$, $ 1 < p \leqslant \infty $, $ \lambda $ Lebesgue measure, though none of its values is an extreme point of the unit ball of $ V$. This shows that a characterization of the extremal elements in $ {L^p}(\lambda ,V)$ for separable $ V$, given by J. A. Johnson, does not hold in general.


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DOI: https://doi.org/10.1090/S0002-9939-1982-0633279-0
Keywords: Bochner $ {L^p}$-space, extreme point, Stonean space
Article copyright: © Copyright 1982 American Mathematical Society