On proximinality and sets of operators. II: Nonexistence of best approximation from the sets of finite rank operators (Q1081776)

The following results are proved: a) for any positive integer \(n\geq 1\), the set \(K_ n(\ell_ 1,c_ 0)\) is not proximinal in \(L(\ell_ 1,c_ 0)\); b) let Q be a compact Hausdorff space ''containing \(Q_ 0''\) (i.e., Q contains an infinite convergent sequence of distinct elements). Then \(K_ n(\ell_ 1,C(Q))\) is not proximinal in \(K(\ell_ 1,C(Q))\). The above results have several implications; among other things, they answer two questions raised by \textit{F. Deutsch, J. Mach} and \textit{K. Saatkamp} [J. Approximation Theory 33, 199-213 (1981; Zbl 0521.41017)].