Positivity

, Volume 22, Issue 1, pp 83–90 | Cite as

Some loose ends on unbounded order convergence

Article

Abstract

The notion of almost everywhere convergence has been generalized to vector lattices as unbounded order convergence, which proves to be a very useful tool in the theory of vector and Banach lattices. In this short note, we establish some new results on unbounded order convergence that tie up some loose ends. In particular, we show that every norm bounded positive increasing net in an order continuous Banach lattice is uo-Cauchy and that every uo-Cauchy net in an order continuous Banach lattice has a uo-limit in the universal completion.

Keywords

Unbounded order convergence Almost everywhere convergence Vector and Banach lattices Universal completion 

Mathematics Subject Classification

46A40 46B42 

Notes

Acknowledgements

The authors thank Dr. Niushan Gao for many valuable discussions and thank the reviewers for carefully reading the paper and providing many suggestions. The first author is grateful to Dr. Niushan Gao for his guidance.

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Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.School of Electrical EngineeringSouthwest Jiaotong UniversityChengduChina
  2. 2.School of MathematicsSouthwest Jiaotong UniversityChengduChina

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