If a Banach space X contains an asymptotically isometric copy of c(0), then
X fails to have weak normal structure. Consequently, if X is an infinite-d
imensional subspace of (c(0), parallel to . parallel to(infinity)), then X
fails to have weak normal structure. Also, every equivalent renorming of c(
0)(Gamma), for Gamma uncountable, fails to have weak normal structure. Ever
y separable Banach space can be equivalently renormed so as not to contain
an asymptotically isometric copy of c(0). Every Banach space with the gener
alized Gossez-Lami Dozo (GGLD) property fails to contain a subspace isomorp
hic to c(0). (C) 1998 Academic Press.