In this paper it is proved
that for each countable ordinal number α ≧ 2 there exists a separable Banach space
X containing a cone P such that, if JX is the canonical map of X into its bidual
X∗∗, then the α-th iterated w∗-sequential closure Kα(JXP) of JXP fails to be
norm-closed in X∗∗. From such spaces there is constructed a separable space W
containing a cone P such that if 2 ≦ β ≦ α, then Kβ(JWf) fails to be normclosed in
W∗∗. Further, there is constructed a (non-separable) space Z containing a
cone P such that if 2 ≦ β < Ω, then Kβ(JZP) fails to be norm-closed in
z ∗∗.
