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Weak compactness in the classes of strengthened \(\sigma\)-order continuous linear functionals of Riesz spaces. (English) Zbl 0778.46010

Let \(L\) be an Archimedean Riesz space (= vector lattice) and let \({\mathcal A}(L)\) be the set of all strengthened \(\sigma\)-order continuous linear functionals on \(L\). It was proved by Veksler that \({\mathcal A}(L)\) consists exactly of the restrictions to \(L\) of all \(\sigma\)-order continuous functionals defined on the \(\sigma\)-Dedekind completion of \(L\). The author gives some characterizations of the weakly compact subsets of \({\mathcal A}(L)\) in terms of the order structure of \(L\). In particular, it is shown that weak relative compactness and order equicontinuity in \({\mathcal A}(L)\) coincide for uniformly complete \(L\). A generalization (Theorem 2.5) of the main results from F. K. Dashiell [Trans. Am. Math. Soc. 266, 397-413 (1981; Zbl 0493.47017)] is also given.

MSC:

46A40 Ordered topological linear spaces, vector lattices
46A20 Duality theory for topological vector spaces
47B60 Linear operators on ordered spaces

Citations:

Zbl 0493.47017
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References:

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