Characterizations of Generalized Proximinal Subspaces in Real Banach Spaces

Abstract

Let X be a real Banach space, C a closed bounded convex subset of X with the origin as an interior point, and $$p_C$$ the Minkowski functional generated by the set C. This paper is concerned with the problem of generalized best approximation with respect to $$p_C$$ . A property $$(\varepsilon _*)$$ concerning a subspace of $$X^*$$ is introduced to characterize generalized proximinal subspaces in X. A set C with feature as above in the space $$l_1$$ of absolutely summable sequences of real numbers and a continuous linear functional f on $$l_1$$ are constructed to show that each point in an open half space determined by the kernel of f admits a generalized best approximation from the kernel but each point in the other open half space does not.