ASYMPTOTICALLY ISOMETRIC COPIES OF lp (1 ≤ p < ∞) AND co IN BANACH SPACES*

Let X be a Banach space. If there exists a quotient space of X which is asymptotically isometric to l1, then X contains complemented asymptotically isometric copies of l1. Every infinite dimensional closed subspace of l1 contains a complemented subspace of l1 which is asymptotically isometric to l1. Let X be a separable Banach space such that X* contains asymptotically isometric copies of lp (1 < p < ∞). Then there exists a quotient space of X which is asymptotically isometric to . Complemented asymptotically isometric copies of co in K(X,Y) and W(X,Y) are discussed. Let X be a Gelfand-Phillips space. If X contains asymptotically isometric copies of co, it has to contain complemented asymptotically isometric copies of co.