ABSTRACT. There has been much recent interest in the problem of approximating in the space of bounded linear operators L(X, Y) from one normed linear space X into another Y by certain subsets U of L(X, Y). In particular, the cases when U = K(X, Y), the compact operators, or K_{N}(X, Y), the rank N operators, have received considerable attention (see, e.g., [7, 10-17]).

This paper represents a further contribution to a solution of the problem: When is K_{N}(X, Y) proximinal in L(X, Y) or in K(X, Y)? The main result of Section 2 (Theorem 2.2) states that K_{N}(X, Y^{∗}) is proximinal in L(X, Y^{∗}) for any normed spaces X and Y.