c © 1997 Universitat de Barcelona Dual action of asymptotically isometric copies of lp(1 ≤ p <∞) and c0

P.N. Dowling and C.J. Lennard proved that if a Banach space contains an asymp-totically isometric copy of l1, then it fails the fixed point property. In this paper, necessary and sufficient conditions for a Banach space to contain an asymptotically isometric copy of lp(1 ≤ p < ∞) or c0 are given by the dual action. In particular, it is shown that a Banach space contains an asymptotically isometric copy of l1 if its dual space contains an isometric copy of l∞, and if a Banach space contains an asymptotically isometric copy of c0, then its dual space contains an asymptotically isometric copy of l1. §1. Preliminaries A Banach space X is said to have an asymptotically isometric copy of l1 [2], if for every sequence (n) (0 < n < 1) decreasing to 0, there exists a norm-one sequence (xn) in X such that