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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Weak compactness in the order dual of a vector lattice


Author: Owen Burkinshaw
Journal: Trans. Amer. Math. Soc. 187 (1974), 183-201
MSC: Primary 46A40
DOI: https://doi.org/10.1090/S0002-9947-1974-0394098-6
MathSciNet review: 0394098
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Abstract: A sequence $ \{ {x_n}\} $ in a vector lattice E will be called an l'-sequence if there exists an x in E such that $ \Sigma _{k = 1}^n\vert{x_k}\vert \leq x$ for all n. Denote the order dual of E by $ {E^b}$. For a set $ A \subset {E^b}$, let $ {\left\Vert \cdot \right\Vert _{{A^ \circ }}}$ denote the Minkowski functional on E defined by its polar $ {A^ \circ }$ in E. A set $ A \subset {E^b}$ will be called equi-l'-continuous on E if $ \lim {\left\Vert {{x_n}} \right\Vert _{{A^ \circ }}} = 0$ for each l'-sequence $ \{ {x_n}\} $ in E.

The main objective of this paper will be to characterize compactness in $ {E^b}$ in terms of the order structure on E and $ {E^b}$. In particular, the relationship of equi-l'-continuity to compactness is studied. §2 extends to $ {E^{\sigma c}}$ the results in Kaplan [8] on vague compactness in $ {E^C}$. Then this is used to study vague convergence of sequences in $ {E^b}$.


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DOI: https://doi.org/10.1090/S0002-9947-1974-0394098-6
Article copyright: © Copyright 1974 American Mathematical Society