The dual space of [formula omitted] and the p-approximation property

Abstract

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 τ p 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 l q which fails to have the p-AP, when q > 2 p / ( p − 2 ) . This subspace is the Davie space in l q (Davie (1973) [5]) which does not have the approximation property. It follows that for every 2 < p < ∞ there exists a Banach space Y p 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 τ p . Finally we introduce the p-compact approximation property ( p-CAP) and show for every 2 < p < ∞ that the Davie space in c 0 fails to have the p-CAP, and also that a variant of the Willis space (Willis (1992) [17]) has the p-CAP, but it fails to have the p-AP.