On proximinality and sets of operators. III. Approximation by finite rank operators on spaces of continuous functions

Abstract

In this case y,, is called a “best approximation” for x from A. If B is a sub- set of X, then 6(B, A)=sup{d(x, A);xeB}, is the deviation of B from A, and u!,(B, X)=inf{G(B, N); N’ IS an n-dimensional subspace of X} is the Kolmogrov n-width of B in X. 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,(X, Y) the set of all operators in L(X, Y) of rank dn. The first serous study of the proximinality of K,(X, Y) in K(X, Y) and L(X, Y) appeared in the paper of Deutsch, Mach, and Saatkamp [2]. This paper was followed by two others, Kamal [4] and Kamal [S], in which several results concerning the proximinality of of K,(X, Y) in K(X, Y) and L(X, Y) were proved. In their paper [2], Deutsch et al. proved that for each integer IZ 2 0, the set K,(c,, c,) is proximinal in L(c,, c,), while in the present paper the following result is proved: Let Q and S be locally com-