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The relations among the notions of various kinds of stability and their applications

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Abstract

First, we prove that a random metric space can be isometrically embedded into a complete random normed module, as an application it is easy to see that the notion of d-\(\sigma \)-stability in a random metric space can be regarded as a special case of the notion of \(\sigma \)-stability in a random normed module; as another application we give the final version of the characterization for a d-\(\sigma \)-stable random metric space to be stably compact. Second, we prove that an \(L^{p}\)-normed \(L^{\infty }\)-module is exactly generated by a complete random normed module so that the gluing property of an \(L^{p}\)-normed \(L^{\infty }\)-module can be derived from the \(\sigma \)-stability of the generating random normed module, as applications the direct relation between module duals and random conjugate spaces are given. Third, we prove that a random normed space is order complete iff it is \((\varepsilon ,\lambda )\)-complete, as an application it is proved that the d-decomposability of an order complete random normed space is exactly its d-\(\sigma \)-stability. Finally, we prove that an equivalence relation on the product space of a nonempty set X and a complete Boolean algebra B is regular iff it can be induced by a B-valued Boolean metric on X, as an application it is proved that a nonempty subset of a Boolean set (X, d) is universally complete iff it is a B-stable set defined by a regular equivalence relation.

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Acknowledgements

This research is supported by the National Natural Science Foundation of China (Grant Nos. 12371141,11971483) and the Provincial Natural Science Foundation of Hunan (Grant No. 2023JJ30642). We would like to thank the reviewers for presenting lots of valuable suggestions which considerably improve the readability of this paper.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Central South University, Changsha, 410083, China

    Tiexin Guo, Xiaohuan Mu & Qiang Tu

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  1. Tiexin Guo
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  2. Xiaohuan Mu
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  3. Qiang Tu
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Correspondence to Tiexin Guo.

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Communicated by Antonio M. Peralta.

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Guo, T., Mu, X. & Tu, Q. The relations among the notions of various kinds of stability and their applications. Banach J. Math. Anal. 18, 42 (2024). https://doi.org/10.1007/s43037-024-00354-w

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