On transitivity of proximinality
A normed linear space E is said to be a P(n) space (n\(\geq 2)\) if \(G^{(P)}\subset M\), \(M^{(P)}\subset E\), \(\dim E/G\leq n\Rightarrow G^{(P)}\subset E,\) where G and M are closed subspaces of E and \(G^{(P)}\subset M\) means that G is a proximinal subspace of M. A normed linear space is called a P space if it is a P(n) space for every \(n\geq 2\), i.e. proximinality is transitive. The characterization of P(n) spaces is given. It is proved that \(C_ 0\) and the incomplete inner product spaces are P spaces. It is also proved that the infinite dimensional C(Q) and \(L^ 1(T,v)\) spaces are not P(2) spaces.
scientific article; zbMATH DE number 4006909
scientific article