Positivity 13 (2009), 21–30
2008 Birkh¨auser Verlag Basel/Switzerland
1385-1292/010021-10, published online August 9, 2008
Characterizations of Riesz spaces
S¸afak Alpay and Zafer Ercan
Abstract. A Riesz space E is said to have b-property if each subset which
is order bounded in E
is order bounded in E. The relationship between
b-property and completeness, being a retract and the absolute weak topology
,E) is studied. Perfect Riesz spaces are characterized in terms of
b-property. It is shown that b-property coincides with the Levi property in
Dedekind complete Frechet lattices.
Mathematics Subject Classiﬁcation (2000). Primary 46A40.
Keywords. Riesz spaces, b-property, locally solid Riesz spaces, Levi property.
Rieszs spaces considered in this note are assumed to have separating order duals.
The order dual of a Riesz space E is denoted by E
denotes the order bidual
of E. Order continuous dual of E is denoted by E
will denote the topological
dual of a topological Riesz space E. It is well known that E
is a band in E
is an ideal of E
. We use without further explanation the basic terminology
and results from the theory of Riesz spaces as set out in [1,2,9,10] and .
Deﬁnition 1. Let E be a Riesz space.
(a) AsubsetA of a Riesz space E is called b-bounded if A is order bounded in
(b) E is said to have b-property if each b-bounded subset of E is order bounded
Although b-boundedness was ﬁrst deﬁned explicitly in  similar ideas were
already considered by others. For example, D. Fremlin had considered subsets
of a Riesz space E that are order bounded in the universal completion of E (,
page 170). Cartwright and Lotz have also used similar ideas in their isomorphic
characterization of M-spaces. They proved that a Banach lattice E is isomorphic