Correction of a generalization of a theorem of Beurling and Livingston
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- by Peter A. Fowler PDF
- Proc. Amer. Math. Soc. 74 (1979), 56-58 Request permission
Correction: Proc. Amer. Math. Soc. 74 (1979), 56-58.
Original Article: Proc. Amer. Math. Soc. 74 (1979), 56-58.
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
- Arne Beurling and A. E. Livingston, A theorem on duality mappings in Banach spaces, Ark. Mat. 4 (1962), 405–411 (1962). MR 145320, DOI 10.1007/BF02591622
- Felix E. Browder, On a theorem of Beurling and Livingston, Canadian J. Math. 17 (1965), 367–372. MR 176320, DOI 10.4153/CJM-1965-037-2
- Felix E. Browder, Multi-valued monotone nonlinear mappings and duality mappings in Banach spaces, Trans. Amer. Math. Soc. 118 (1965), 338–351. MR 180884, DOI 10.1090/S0002-9947-1965-0180884-9
- Robert C. James, Linear functionals as differentials of a norm, Math. Mag. 24 (1951), 237–244. MR 44023, DOI 10.2307/3029073
Additional Information
- © Copyright 1979 American Mathematical Society
- 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
- MathSciNet review: 521873