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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
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.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1979-0521873-7
PII: S 0002-9939(1979)0521873-7
Article copyright: © Copyright 1979 American Mathematical Society