Proximinal subspaces of finite codimension in direct sum spaces

We give a necessary and sufficient condition for proximinality of a closed subspace of finite codimension in c₀-direct sum of Banach spaces.     

Some remarks on proximinality in higher dual spaces

In this paper we consider proximinality questions for higher ordered dual spaces. We show that for a finite dimensional uniformly convex space X, the space C(K,X) is proximinal in all the duals of even order. For any family of uniformly convex Banach spaces {Xα}_{α∈Γ} we show that any finite co-dimensional proximinal subspace of X=c_0⊕Xα is strongly proximinal in all the duals of even order of X.     

On proximinality of convex sets in superspaces

In this paper, we show that a closed convex subset C of a Banach space is strongly proximinal (proximinal, resp.) in every Banach space isometrically containing it if and only if C is locally (weakly, resp.) compact. As a consequence, it is proved that local compactness of C is also equivalent to that for every Banach space Y isometrically containing it, the metric projection from Y to C is nonempty set-valued and upper semi-continuous.     

On the proximinality of the unit ball of proximinal subspaces in Banach spaces: A counterexample

A known, and easy to establish, fact in Best Approximation Theory is that, if the unit ball of a subspace of a Banach space is proximinal in , then itself is proximinal in . We are concerned in this article with the reverse implication, as the knowledge of whether the unit ball is proximinal or not is useful in obtaining information about other problems. We show, by constructing a counterexample, that the answer is negative in general.

     

On a Garkavi Type Theorem for M-Ideals of Finite Codimension

A well-known result of Garkavi asserts that any proximinal subspace of finite codimension is an intersection of proximinal hyperplanes. In this article we investigate Banach spaces in which every M-ideal of finite codimension, is an intersection of M-ideals of codimension one. We show that for spaces that have M-ideals of codimension one, this property is preserved under c ₀-direct sums. We also consider this question for factor reflexive subspaces and M-summands of finite codimension.