On a sufficient condition for proximity

Lau, Ka Sing

doi:10.1090/S0002-9947-1979-0531983-0

On a sufficient condition for proximity
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Trans. Amer. Math. Soc. 251 (1979), 343-356 Request permission

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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