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 2 < p < ∞, there exists a subspace of lq which fails to have the p-AP, when q > 2p/(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 < p < ∞ there exists a Banach space Yp such that it has the p-AP, but its dual space Y∗ p fails to have the p-AP. We study the relation of the p-AP with the denseness of finite rank operators in the topology τp. Finally we introduce the p-compact approximation property (p-CAP) and show for every 2