Let X be a Banach space and (Omega, Sigma, mu) be a finite measure space. I
t is shown that if 1 less than or equal to p < infinity (resp. 1 < p < infi
nity), the Bochner space L-p(mu, X) contains asymptotically isometric copie
s of c(u) (resp. l(1)) if and only if X does. (C) 2001 Academic Press.