, Volume 176, Issue 1, pp 209-220  Weak convergence in the dual
of weak Lp  Purchase on Springer.com  $39.95 / €34.95 / £29.95*
Rent the article at a discount  Rent now  * Final gross prices
may vary according to local VAT.  Get Access   We consider a
Dedekind σ-complete Banach lattice E whose dual is weakly
sequentially complete. Suppose that E has a positive element u
and a family of positive operators $ \mathcal{G} $ such that
1. each T′, T $ \mathcal{G} $ , is a lattice homomorphism    2.
$ \cup _{T \in \mathcal{G}} $ [−u, u] contains the unit ball of
E    3. for any sequence (x n ) ⊂ [0, u] of pairwise disjoint
elements and for any sequence (T n ) ⊂ $ \mathcal{G} $ the
sequence (T n x n ) is majorized in E.  We show that such a
space is a Grothendieck space, i.e., in the dual every weak*
convergent sequence converges weakly (Theorem 1). We prove that
Weak L p on a real interval satisfies the conditions above if 1
< p < ∞ (Theorem 2). Then we show that every Weak L p space is a
Grothendieck space (The