%0 Journal Article
%T Coproximinality in the Space of Bounded Functions
%A Eyad Abu-Sirhan
%A Zuhier Altawallbeh
%J International Journal of Mathematics and Mathematical Sciences
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/196391
%X As a counterpart to best approximation in Banach spaces, the best coapproximation was introduced by Franchetti and Furi (1972). In this paper, we shall consider the relation between coproximinality of a nonempty subset of a Banach space and of in . 1. Introduction Let be a nonempty subset of a Banach space . We say that is proximinal in if each there corresponds at least to one point such that Recently, another kind of approximation from has been introduced by Franchetti and Furi [1] who have considered those elements (if any) satisfying Such an element is called best coapproximant of in . The set of all such elements, satisfying above inequality, is denoted by . The subset is called coproximinal in if is nonempty for any . If is singleton for any , then is called coChebyshev; see [2¨C4]. It is clear that is convex if is convex and closed. The kernal of is the set defined by Many results in the theory of best coapproximation have appeared since Franchetti and FurriĄ¯s paper, 1972. These results are largely concerned with the question of existence and uniqueness of best coapproximation (see for example [5¨C8]). Let and denote the Banach spaces of all bounded (resp., continuous) functions from a topological space into a Banach space and let be a closed subset of . It should be remarked that if is a compact space, then is a subspace of . In this paper, we discuss the coproximinality of and in and , respectively. For uniqueness and existence of best coapproximation in , see [7]. Definition 1. Let be a coproximinal subset of a Banach space . A map which associates with each element in one of its best coapproximation in (i.e., for all ) is called a coproximity map. Remark 2. Let be a coproximinal subset of a Banach space . We state some basic properties of a proximity map .(1)If is coChebyshev, [5], then(a) , for any .(b) , for any scalar and . (i.e., is homogeneous).(c) , for any and .(2)If is a subspace of , then(a) , for any and .(b) , for all . (set and in (a) and take into account that ).(c) is continuous at .(d)If is linear, then is continuous. Theorem 3 (see [3]). If is coproximinal hyperplane or -dimensional subspace of a Banach space , then has a continuous coproximity map. Lemma 4 (see [3]). Let be a coproximinal subspace of . Then the following are equivalent.(1) has linear coproximity map.(2) contains a closed subspace such that . Moreover, if (2) holds, then the linear coproximity map for can be defined by , , and . Definition 5. Let be a Banach space. Consider the following.(a)A subspace of is called one complemented in if there is a closed
%U http://www.hindawi.com/journals/ijmms/2014/196391/