An order topology in ordered topological vector spaces

Carter, Lyne H.

doi:10.1090/S0002-9947-1976-0390704-2

An order topology in ordered topological vector spaces
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by Lyne H. Carter

Trans. Amer. Math. Soc. 216 (1976), 131-144

DOI: https://doi.org/10.1090/S0002-9947-1976-0390704-2

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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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