Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On a sufficient condition for proximity

Author: Ka Sing Lau
Journal: Trans. Amer. Math. Soc. 251 (1979), 343-356
MSC: Primary 46B99; Secondary 41A65, 47D15
MathSciNet review: 531983
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Abstract: A closed subspace M in a Banach space X is called U-proximinal if it satisfies: $ (1 + \rho )S \cap (S + M) \subseteq S + \varepsilon (\rho )(S \cap M)$, for some positive valued function $ \varepsilon (\rho )$, $ \rho > 0$, and $ \varepsilon (\rho ) \to 0$ as $ \rho\, \to\, 0$, where S is the closed unit ball of X. One of the important properties of this class of subspaces is that the metric projections are continuous. We show that many interesting subspaces are U-proximinal, for example, the subspaces with the 2-ball property (semi M-ideals) and certain subspaces of compact operators in the spaces of bounded linear operators.

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Keywords: Compact operators, measurable functions, M-ideals, proximity, uniformly convex
Article copyright: © Copyright 1979 American Mathematical Society