ASYMPTOTICALLY ISOMETRIC COPIES OF l_P (1≤P<∞) AND c_0 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 lq (1/p 1/q=1).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.