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Proceedings of the American Mathematical Society

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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
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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  • [1] E. Behrends et al., $ {L^p}$-structure in real Banach spaces, Lecture Notes in Math., vol. 613, Springer-Verlag, Berlin, 1977. MR 0626051 (58:30082)
  • [2] R. M. Blumenthal, J. Lindenstrauss and R. R. Phelps, Extreme operators into $ C(K)$, Pacific J. Math. 15 (1965), 747-756. MR 0209862 (35:758)
  • [3] J. A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), 396-414. MR 1501880
  • [4] J. Diestel and J. J. Uhl, Jr., Vector measures, Math. Surveys, No. 15, Amer. Math. Soc., Providence, R. I., 1977. MR 0453964 (56:12216)
  • [5] J. A. Johnson, Extreme measurable selections, Proc. Amer. Math. Soc. 44 (1974), 107-112. MR 0341068 (49:5818)
  • [6] K. Sundaresan, Extreme points of the unit cell in Lebesgue-Bochner function spaces, Colloq. Math. 22 (1970), 111-119. MR 0276753 (43:2493)

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Keywords: Bochner $ {L^p}$-space, extreme point, Stonean space
Article copyright: © Copyright 1982 American Mathematical Society

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