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 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 (Theorem 3).