1964
DOI: 10.1215/ijm/1256059459
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Partially ordered linear spaces and locally convex linear topological spaces

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Cited by 36 publications
(23 citation statements)
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“…Following [42,18,46,34,26], a net (x α ) in a vector lattice X is said to converge in unbounded order (uo-converge for short) to x ∈ X, written as x α uo − → x, if |x α − x| ∧ y o − → 0 for any y ∈ X + ; (x α ) is said to be uo-Cauchy if the "double" net (x α − x β ) (α,β) uoconverges to zero. It is easily seen that uo-convergence (respectively, uo-Cauchy) coincides with order convergence (respectively, o-Cauchy) for order bounded nets.…”
Section: Unbounded Order Convergence and Regular Sublatticesmentioning
confidence: 99%
“…Following [42,18,46,34,26], a net (x α ) in a vector lattice X is said to converge in unbounded order (uo-converge for short) to x ∈ X, written as x α uo − → x, if |x α − x| ∧ y o − → 0 for any y ∈ X + ; (x α ) is said to be uo-Cauchy if the "double" net (x α − x β ) (α,β) uoconverges to zero. It is easily seen that uo-convergence (respectively, uo-Cauchy) coincides with order convergence (respectively, o-Cauchy) for order bounded nets.…”
Section: Unbounded Order Convergence and Regular Sublatticesmentioning
confidence: 99%
“…It follows that order intervals are uo-and un-closed. For sequences in L p (µ), where 1 p < ∞ and µ is a finite measure, it is easy to see that uo-convergence agrees with convergence almost everywhere, see, e.g., [DeM64,Example 2]. Under the same assumptions, un-convergence agrees with convergence in measure, see [Tro04,Example 23].…”
Section: Introductionmentioning
confidence: 99%
“…The concept of unbounded order convergence under the name of individual convergence was first considered in [13] and " uo -convergence" was initially proposed in [6]. Recently, several papers about uo -convergence in Banach lattices have been published; see [3-5, 8-10, 16] for more details on these results.…”
Section: Introductionmentioning
confidence: 99%