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It is shown that the spaces (£(S)E) ioo^ are Grothendieck spaces for a class of Banach lattices E which includes the Orlicz spaces with weakly sequentially complete duals. A Banach space is said to be a Grothendieck space if weak* and weak sequential convergence coincide in the dual. The simplest nontrivial example of a Grothendieck space is Z°°. In [7], the question of when the space (E © £)i (r) is Grothendieck if 2 i|| defines a finitely additive measure on Y; consequently, we may identify px* with an element of/°°(r)'. LEMMA 1. If(x[) is a positive weak* null sequence in F', then (px') is relatively weakly compact in l°°(r)'. PROOF. Let pi = px>. If (pi) is not relatively weakly compact, then there exist a partition (A?) of T and e > 0 such that Pi(Ai) > £ for all i. By definition of pi, there is a positive normalized sequence (x?) C F such that x?xa? = 0 and (x?,x?) > £ for all ». Let x = sup?x?. Then ||x|| = 1 and (x,x?) > e for all i, contrary to the fact that (x?) is weak* null. THEOREM 2. Let E be a Banach lattice with positive cone E+. Suppose there exist a function r : E+ —► [0,00] and a positive real number M with the following properties: (1) r(0) = 0; (2) ||x|| sup, x? exists and || sup,-Xj || <M. Received by the editors February 2, 1987 and, in revised form, May 21, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 46E30, 46B30. ©1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page

Weak* convergence in higher duals of Orlicz spaces