Let Z be a Banach space and G be a closed subspace of Z. For f1,f2 \in Z, the distance from f1,f2 to G is defined by d(f1,f2,G) = \underset{f \in G}{\inf} max {||f1-f||, ||f2-f||}. An element g\ast \in G satisfying max {||f1-g\ast ||, || f2-g\ast ||} = \underset{f \in G}{\inf } max {|| f1-f||, ||f2-f||} is called a best simultaneous approximation for f1,f2 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).

Let Z be a Banach space and G be a closed subspace of Z. For f1,f2 \in Z, the distance from f1,f2 to G is defined by d(f1,f2,G) = \underset{f \in G}{\inf} max {||f1-f||, ||f2-f||}. An element g\ast \in G satisfying max {||f1-g\ast ||, || f2-g\ast ||} = \underset{f \in G}{\inf } max {|| f1-f||, ||f2-f||} is called a best simultaneous approximation for f1,f2 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).