Asymptotically isometric copies of c0 and l1 in certain Banach spaces

Abstract

If X is a separable Banach space, then X ∗ contains an asymptotically isometric copy of l 1 if and only if there exists a quotient space of X which is asymptotically isometric to c 0. If X is an infinite-dimensional normed linear space and Y is any Banach space containing an asymptotically isometric copy of c 0, 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 c 0, then K w ∗ (X ∗,Y) contains a complemented asymptotically isometric copy of c 0.