Best simultaneous approximation in function and operator spaces

Let Z be a Banach space and G be a closed subspace of Z. For f_1,f_2 in Z, the distance from f_1,f_2 to G is defined by d(f_1,f_2,G) = underset{f in G}{inf} max {||f_1-f||, ||f_2-f||}. An element g^{ast} in G satisfying max {||f_1-g^{ast }||, || f_2-g^{ast }||} = underset{f in G}{inf } max {|| f_1-f||, ||f_2-f||} is called a best simultaneous approximation for f_1,f_2 from G. In this paper, we study the problem of best simultananeous approximation in the space of all continuous X-valued functions on a compact Hausdorff space S; C(S,X), and the space of all Bounded linear operators from a Banach space X into a Banach space Y; L(X,Y).