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 is a complete metrizable topological vector lattice, a sequence converges for the topology 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 is locally convex, Namioka [11, Theorem 5.4] has shown that coincides with the order bound topology 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 it is shown for the nonnecessarily local convex topology that and when is locally convex.

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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