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Correction of a generalization of a theorem of Beurling and Livingston


Author: Peter A. Fowler
Journal: Proc. Amer. Math. Soc. 74 (1979), 56-58
MSC: Primary 46B10; Secondary 47H99
DOI: https://doi.org/10.1090/S0002-9939-1979-0521873-7
Correction: Proc. Amer. Math. Soc. 74 (1979), 56-58.
Original Article: Proc. Amer. Math. Soc. 74 (1979), 56-58.
MathSciNet review: 521873
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Abstract: A generalization of a Riesz-Fischer theorem proved by Beurling and Livingston for smooth uniformly convex Banach spaces also holds for smooth, strictly convex, reflexive Banach spaces. Theorem. Let B be a smooth, strictly convex, reflexive Banach space. Let $ L:B \to {B^\ast}$ be a duality map, C a closed subspace of $ B,h \in B,k \in {B^\ast}$. Then $ T(C + h) \cap ({C^ \bot } + k)$ is a single point. A two-dimensional counterexample shows that $ T(C + h) \cap ({C^ \bot } + k) = \emptyset $ is possible if B is not smooth, contrary to the claim of Theorem 4 of Browder, On a theorem of Beurling and Livingston, Canad. J. Math. 17 (1965), 367-372.


References [Enhancements On Off] (What's this?)

  • [1] A. Beurling and A. E. Livingston, A theorem on duality mappings in Banach spaces, Ark. Mat. 4 (1961), 405-411. MR 0145320 (26:2851)
  • [2] F. E. Browder, On a theorem of Beurling and Livingston, Canad. J. Math. 17 (1965), 367-372. MR 0176320 (31:595)
  • [3] -, Multivalued monotone nonlinear mappings and duality mappings in Banach space, Trans. Amer. Math. Soc. 118 (1965), 338-351. MR 0180884 (31:5114)
  • [4] R. C. James, Linear functionals as differentials of a norm, Math. Mag. 24 (1951), 237-244. MR 0044023 (13:356c)

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DOI: https://doi.org/10.1090/S0002-9939-1979-0521873-7
Article copyright: © Copyright 1979 American Mathematical Society

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