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Transactions of the American Mathematical Society

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

 

 

An order topology in ordered topological vector spaces


Author: Lyne H. Carter
Journal: Trans. Amer. Math. Soc. 216 (1976), 131-144
MSC: Primary 46A40
MathSciNet review: 0390704
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Abstract: An order topology $ \Omega $ that can be defined on any partially-ordered space has as its closed sets those that contain the (o)-limits of all their (o)-convergent nets. In this paper we study the situation in which a topological vector space with a Schauder basis is ordered by the basis cone. In a Fréchet space $ (E,\tau )$, we obtain necessary and sufficient conditions both for $ \tau \subset \Omega $ and for $ \tau = \Omega $. Characterizations of (o)- and $ \Omega $-convergence and of $ \Omega $-closed sets are obtained. The equality of the order topology with the strong topology in certain dual Banach spaces is related to weak sequential completeness through the concept of a shrinking basis.


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DOI: https://doi.org/10.1090/S0002-9947-1976-0390704-2
Keywords: (o)-convergence, Schauder basis, unconditional basis, shrinking basis, basis cone, normal cone, barrelled space, Fréchet space, Banach space
Article copyright: © Copyright 1976 American Mathematical Society