, Volume 176, Issue 1, pp 209-220

Weak convergence in the dual of weak Lp

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Abstract

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 (Theorem 3).

The author is indebted to the referee for his detailed comments and valuable suggestions