The dual space of (L(X,Y),τp) and the p-approximation property

We establish a representation of the dual space of L(X,Y), the space of bounded linear operators from a Banach space X into a Banach space Y, endowed with the topology τp of uniform convergence on p-compact subsets of X. We apply this representation and solve the duality problem for the p-approximation property (p-AP), that is, if the dual space X∗ has the p-AP, then so does X. However, the converse does not hold in general. We show that given 22p/(p−2). This subspace is the Davie space in lq (Davie (1973) [5], ) which does not have the approximation property. It follows that for every 2