HAHN-BANACH THEOREM FOR THE COMPACT CONVERGENCE TOPOLOGY AND APPLICATIONS TO APPROXIMATION PROPERTIES

Abstract

Let X and Y be Banach spaces. We show that if X* or Y* has the Radon-Nikodym property and y is a subspace of K-w*(w) (X*, Y), the space of weak* to weak continuous compact operators from X* into Y, then for every phi is an element of y* and epsilon > 0 there exists a phi(epsilon) is an element of (B(X*, Y), tau)*, where tau is the compact convergence topology on the space B(X*, Y) of bounded linear operators, such that phi(epsilon)(S) = phi(S) for all S is an element of Y and parallel to phi(epsilon)parallel to <= parallel to phi parallel to vertical bar c. From this result we obtain various applications.