Positivity 7: 135–139, 2003. © 2003 Kluwer Academic Publishers. Printed in the Netherlands. 135 On Property (b) of Vector Lattices S. ALPAY 1 , B. ALTIN 2 and C. TONYALI 2 1 Department of Mathematics, Middle East Technical University, Ankara, Turkey; 2 Department of Mathematics, Gazi University, Besevler, Ankara, Turkey Abstract. A boundedness property is introduced and characterizations of this property are given. 1. Introduction The Riesz spaces considered in this note are assumed to have separating order duals. In all undefined terminology concerning Riesz spaces we will adhere to [2] and [4]. We start by identifying a boundedness property. DEFINITION 1.1. A subset A of a Riesz space E is called b-order bounded in E if it is order bounded in (E ∼ ) ∼ . A Riesz space is said to have property (b) if A ⊂ E is order bounded whenever A is order bounded in (E ∼ ) ∼ . A Riesz space E has property (b) if and only if for each net {x α } in E with 0  x α ↑ ˆ x for some ˆ x in (E ∼ ) ∼ , {x α } is order bounded in E. EXAMPLE 1.2. Every perfect Riesz space and therefore every order dual has property (b). Every reflexive Banach lattice has property (b). Every KB space has property (b) and if (E ∼ ) ∼ is retractable on E then E has (b). On the other hand, by considering A ={e n } in c 0 , we see that c 0 does not have property (b). Let E, F be Banach lattices and F be an KB -space. Then L b (E,F) has property (b). For an arbitrary compact Hausdorff space K,C(K) has property (b). The weak Fatou property for directed sets (cf. [4]) of normed Riesz spaces im- plies the (b)-property. The Levi property and Zaanen’s B-property also imply the (b)-property in Frechet lattices. However, C[0, 1], with the supremum norm, has neither the Levi nor the B-property. Does property (b) pass to subspaces? Not, in general. For example each infinite dimensional closed Riesz subspace F of a C(K) space contains a (lattice) copy of c 0 . In ℓ p spaces (1  p< ∞) every closed sublattice is the range of a positive (contractive) projection. Hence closed sublattices of ℓ p inherit property (b) [1]. On the other hand, every projection band in a Riesz space with property (b) has property (b).