### Proximinality of certain subspaces of \$C_b(S;E)\$

Joao B. Prolla, Ary O. Chiacchio
1989 Rocky Mountain Journal of Mathematics
Throughout this paper, 5 is a completely regular HausdorfF space and E is a Banach space. The vector space of all continuous and bounded functions / : S -• E, denoted by Cb(S;E), is equipped with the supnorm ||/|| = sup{||/(x)||;xG5}. Recall that a closed subpsace V of a Banach space E is said to be proximinal if every a G E admits a best approximant from V', i.e., a point v G V for which The set of best approximants to a from V is denoted by Py(a), and the set-valued mapping a -• Py(a) is
more » ... d the metric projection. If V is proximinal, then a -• Py{a) # 0 for every a G E. If Py(a) is a singleton for each a G £, then V is called a Chebyshev subspace of £. If V is a proximinal subspace of E, then a map s : £ -> V such that s(a) belongs to Pv(a), for each a G -E 1 , is called a metric selection or a proximity map for V. The following notations are standard and will be used throughout this paper. If a G E and r > 0, B(a;r) = {v G E;\\v -a\\ < r} and B(a;r) -{v G E : \\v -a\\ < r}. For any s G 5, the bounded linear operator <5 S : Cb(S;E) -• £ is defined by é s (/) = /(s), for all / € Cfe(5; £). If W is a closed vector subspace of Cb{S; E), then 6 S \W denotes the restriction of S 8 to W. Notice that 0< ||Ó S |W|| < 1. Given a proximinal subspace V of a Banach space E, then clearly Cb(S; V) is a closed subspace of Cb(S; E). In this paper we shall study the following questions. QUESTION 1. Under what assumptions is Cb{S;V) proximinal in C b {S;E)?