Asymptotically isometric copies of l(infinity) in Banach spaces and a theorem of Bessaga and Pelczynski

We introduce the notion of a Banach space containing an asymptotically isom etric copy of l(infinity). A well known result of Bessaga and Pelczynski st ates a Banach space X contains a complemented isomorphic copy of l(1) if an d only if X* contains an isomorphic copy of c(0) if and only if X* contains an isomorphic copy of l(infinity). We prove an asymptotically isometric an alogue of this result.