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Comparison of two types of order convergence with topological convergence in an ordered topological vector space


Authors: Roger W. May and Charles W. McArthur
Journal: Proc. Amer. Math. Soc. 63 (1977), 49-55
MSC: Primary 46A40
DOI: https://doi.org/10.1090/S0002-9939-1977-0438078-9
MathSciNet review: 0438078
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Abstract: Birkhoff and Peressini proved that if $ (X,\mathcal{T})$ is a complete metrizable topological vector lattice, a sequence converges for the topology $ \mathcal{T}$ iff the sequence relatively uniformly star converges. The above assumption of lattice structure is unnecessary. A necessary and sufficient condition for the conclusion is that the positive cone be closed, normal, and generating. If, moreover, the space $ (X,\mathcal{T})$ is locally convex, Namioka [11, Theorem 5.4] has shown that $ \mathcal{T}$ coincides with the order bound topology $ {\mathcal{T}_b}$ and Gordon [4, Corollary, p. 423] (assuming lattice structure and local convexity) shows that metric convergence coincides with relative uniform star convergence. Omitting the assumptions of lattice structure and local convexity of $ (X,\mathcal{T})$ it is shown for the nonnecessarily local convex topology $ {\mathcal{T}_{{\text{ru}}}}$ that $ {\mathcal{T}_{\text{b}}} \subset {\mathcal{T}_{{\text{ru}}}} = \mathcal{T}$ and $ {\mathcal{T}_{\text{b}}} = {\mathcal{T}_{{\text{ru}}}} = \mathcal{T}$ when $ (X,\mathcal{T})$ is locally convex.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1977-0438078-9
Keywords: Relative uniform convergence, order convergence, normal cone, generating cone, relatively uniform topology, order topology
Article copyright: © Copyright 1977 American Mathematical Society

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