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Some loose ends on unbounded order convergence

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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.

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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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  1. School of Electrical Engineering, Southwest Jiaotong University, Chengdu, 610000, Sichuan, China

    Hui Li

  2. School of Mathematics, Southwest Jiaotong University, Chengdu, 610000, Sichuan, China

    Zili Chen

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  1. Hui Li
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  2. Zili Chen
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Correspondence to Hui Li.

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Li, H., Chen, Z. Some loose ends on unbounded order convergence. Positivity 22, 83–90 (2018). https://doi.org/10.1007/s11117-017-0501-1

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  • DOI: https://doi.org/10.1007/s11117-017-0501-1

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