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

Release time:2024-10-25

Impact Factor:1.1

DOI number:10.1007/s43037-024-00354-w

Journal:Banach J. Math. Anal.

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-σ-stability in a random metric space can be regarded as a special case of the notion of σ-stability in a random normed module; as another application we give the final version of the characterization for a d-σ-stable random metric space to be stably compact. Second, we prove that an Lp-normed L∞-module is exactly generated by a complete random normed module so that the gluing property of an Lp-normed L∞- module can be derived from the σ-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 (ε, λ)-complete, as an application it is proved that the d-decomposability of an order complete random normed space is exactly its d-σ-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.

Co-author:Xiaohuan Mu, Qiang Tu

First Author:Tiexin Guo

Indexed by:Journal paper

Document Code:42

Discipline:Natural Science

First-Level Discipline:Mathematics

Document Type:J

Volume:18

Translation or Not:no

Date of Publication:2024-05-24

Included Journals:SCI