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

May, Roger W.; McArthur, Charles W.

doi:10.1090/S0002-9939-1977-0438078-9

Comparison of two types of order convergence with topological convergence in an ordered topological vector space
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by Roger W. May and Charles W. McArthur PDF

Proc. Amer. Math. Soc. 63 (1977), 49-55 Request permission

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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Comparison of order topologies with the topology of an ordered topological vector space

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