Characterizations of Generalized Proximinal Subspaces in Real Banach Spaces

Let X be a real Banach space, C a closed bounded convex subset of X with the origin as an interior point, and pC the Minkowski functional generated by the set C. This paper is concerned with the problem of generalized best approximation with respect to pC. A property (epsilon(*)) 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.