Duality for unbounded order convergence and applications

Gao, Niushan; Leung, Denny H.; Xanthos, Foivos

doi:10.1007/s11117-017-0539-0

Cited by 27 publications

(32 citation statements)

References 8 publications

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“…An easy application of Theorem 2.4 is as follows. It is known from [3] that (L p ) ∼ uo = L q for 1 < p ≤ ∞ and p −1 + q −1 = 1 and (L 1 ) ∼ uo = {0}. Thus the relative L 0 -topology is locally convex on the unit ball of L p , 1 < p ≤ ∞, but not on the unit ball of L 1 .…”

Section: Resultsmentioning

confidence: 99%

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A local Hahn–Banach theorem and its applications

2019

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An important consequence of the Hahn-Banach Theorem says that on any locally convex Hausdorff topological space X, there are sufficiently many continuous linear functionals to separate points of X. In the paper, we establish a "local" version of this theorem. The result is applied to study the uo-dual of a Banach lattice that was recently introduced in [3]. We also provide a simplified approach to the measure-free characterization of uniform integrability established in [8].

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Section: Resultsmentioning

confidence: 99%

“…[3,Theorem 2.3] characterizes X ∼ uo as the order continuous part of X ∼ n . We refer to [4] and [3] for facts about uo-convergence and uo-dual. The uo-convergence has garnered interest as the natural generalization of a.s.-convergence.…”

Section: Resultsmentioning

confidence: 99%

A local Hahn–Banach theorem and its applications

2019

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“…Troitsky, and F. Xanthos. The sharpest result is [GLX,Proposition 3.1]; it states that a Banach lattice whose order continuous dual separates points is boundedly uo-complete iff it is monotonically complete. In this section, we remove the restriction on the order continuous dual.…”

Section: Boundedly Uo-complete Banach Latticesmentioning

confidence: 99%

Completeness of unbounded convergences

Taylor

2018

Proc. Amer. Math. Soc.

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Abstract. As a generalization of almost everywhere convergence to vector lattices, unbounded order convergence has garnered much attention. The concept of boundedly uo-complete Banach lattices was introduced by N. Gao and F. Xanthos, and has been studied in recent papers by D. Leung, V.G. Troitsky, and the aforementioned authors. We will prove that a Banach lattice is boundedly uo-complete iff it is monotonically complete. Afterwards, we study completeness-type properties of minimal topologies; minimal topologies are exactly the Hausdorff locally solid topologies in which uo-convergence implies topological convergence.

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“…For general results on uo-convergence and its unbounded norm version we refer the reader to [GX14,Gao14,KMT17,GTX,LC]. For applications of uoconvergence and its techniques to Mathematical finance we refer the reader to [GLX,GLX2].…”

Section: Introductionmentioning

confidence: 99%

The countable sup property for lattices of continuous functions

Kandić

Vavpetič

2018

Journal of Mathematical Analysis and Applications

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In this paper we find sufficient and necessary conditions under which vector lattice C(X) and its sublattices C b (X), C 0 (X) and C c (X) have the countable sup property. It turns out that the countable sup property is tightly connected to the countable chain condition of the underlying topological space X. We also consider the countable sup property of C(X × Y ). Even when both C(X) and C(Y ) have the countable sup property it is possible that C(X × Y ) fails to have it. For this construction one needs to assume the continuum hypothesis. In general, we present a positive result in this direction and also address the question when C( λ∈Λ X λ ) has the countable sup property. Our results can be understood as vector lattice theoretical versions of results regarding products of spaces satisfying the countable chain condition. We also present new results for general vector lattices that are of an independent interest.2010 Mathematics Subject Classification. 46B42, 06E10, 46A40, 46E25.

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Duality for unbounded order convergence and applications

Cited by 27 publications

References 8 publications

A local Hahn–Banach theorem and its applications

A local Hahn–Banach theorem and its applications

Completeness of unbounded convergences

The countable sup property for lattices of continuous functions

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