ASYMPTOTICALLY ISOMETRIC COPIES OF In （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 l^1, then X contains complemented asymptotically isometric copies of l^1. 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.