On a sufficient condition for proximity

Abstract

A closed subspace M in a Banach space X is called U-proximinal if it satisfies: ( 1 + ρ ) S ∩ ( S + M ) ⊆ S + ε ( ρ ) ( S ∩ M ) (1 + \rho )S \cap (S + M) \subseteq S + \varepsilon (\rho )(S \cap M) , for some positive valued function ε ( ρ ) \varepsilon (\rho ) , ρ > 0 \rho > 0 , and ε ( ρ ) → 0 \varepsilon (\rho ) \to 0 as ρ → 0 \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.