The duality of some approximation properties (Q1891329)

A subset \(A\) of a normed linear space \(X\) is said to be proximinal if for each \(x\in X\), the distance \(d(x,A)\) is attained i.e. there is some \(y\in A\) such that \(|x-y |= d(x,A)\). If \(X\) and \(Y\) are two normed linear spaces, then \(L(X,Y)\) is defined to be the Banach space of all bounded linear operators from \(X\) to \(Y\), and \(K(X, Y)\) is the subspace of \(L(X, Y)\) consisting of all compact linear operators. Several results on the proximinality of \(K(X,Y)\) in \(L(X,Y)\) are known in the literature. The author uses the relation between the proximinality properties of the Banach space and its dual to give a simple unified proof to these results and also proves some new results on the proximinality of \(K(X,Y)\) in \(L(X,Y)\). He also characterizes all those spaces \(X\) for which \(K(\ell_1 (Q), X)\) (resp. \(K(X, B(Q))\) is proximinal in \(L(\ell_1 (Q) X)\) (resp. \(L(X, B(Q))\)), where \(Q\) is an arbitrary set, \(\ell_1 (Q)\) is the Banach space of all ral-valued functions \(f: Q\to \mathbb{R}\) satisfying \(\sum_{q\in Q} |f(q)|<\infty\) equipped with the norm \(|f|=\sum_{q\in Q} |f(q) |\), \(B(Q,X)\) is the space of all bounded functions from \(Q\) to \(X\) and when \(X\) is the set of real numbers then \(B(Q, X)\) is denoted by \(B(Q)\). Main result: Let \(Q\) be an arbitrary set and \(X\) a Banach space, then (i) \(K(X, B(Q))\) is proximinal in \(L(X, B(Q))\) if and only if for each \(f\in B(Q, X^*)\), \(K(f(Q))\) is attained, (ii) \(K(\ell_1 (Q), X)\) is proximinal in \(L(\ell_1 (Q), X)\) if and only if for each \(f\in B(Q, X^*)\), \(K(f(Q))\) is attained. (Here for any bounded subset \(A\) of \(X\), \(K(A)= \inf\{ \delta (A,K)\): \(K\) is a relatively compact subset of \(X\), \(\delta (A, K)\equiv \sup\{ d(y,K)\): \(y\in A\})\).