Asymptotically isometric copies of c0 and l1 in certain Banach spaces ✩

If X is a separable Banach space, then X∗ contains an asymptotically isometric copy of l1 if and only if there exists a quotient space of X which is asymptotically isometric to c0. If X is an infinitedimensional normed linear space and Y is any Banach space containing an asymptotically isometric copy of c0, then L(X,Y) contains an isometric copy of l∞. If X and Y are two infinite-dimensional Banach spaces and Y contains an asymptotically isometric copy of c0, then Kw∗(X∗,Y) contains a complemented asymptotically isometric copy of c0.  2003 Elsevier Inc. All rights reserved.