On proximinality and sets of operators. I. Best approximation by finite rank operators (Q1078453)

If X and Y are normed linear spaces, then L(X,Y) denotes the set of all bounded linear operators from X to Y, K(X,Y) the set of all compact operators in L(X,Y) and \(K_ n(X,Y)\) the set of all operators of rank \(\leq n\) in L(X,Y). This paper contains a study for the proximinality of \(K_ n(X,Y)\) in L(X,Y) and K(X,Y). Among the results proved, we quote the following: \(K_ n(X,C_ 0(Q))\) is proximinal in \(L(X,C_ 0(Q))\) if \(X^*\) is uniformly convex and Q is a locally compact Hausdorff space. If \(\dim (X)<\infty\), then \(K_ n(X,C_ 0(Q))\) is proximinal in \(K(X,C_ 0(Q))\) for each locally compact Hausdorff space Q iff the metric projection from \(X^*\) onto any of its n-dimensional subspace has a continuous selection. These theorems give solutions to some problems in this area [see \textit{F. Deutsch}, \textit{J. Mach} and \textit{K. Saatkamp}, J. Approximation Theory 33, 199-213 (1981; Zbl 0521.41017)].