The relations among the notions of various kinds of stability and their applications
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| Format: | Preprint |
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2024
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| _version_ | 1866914665868558336 |
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| author | Guo, Tiexin Mu, Xiaohuan Tu, Qiang |
| author_facet | Guo, Tiexin Mu, Xiaohuan Tu, Qiang |
| contents | First, we prove that a random metric space can be isometrically embedded into a complete random normed module, as an application of which, it is easy to see that the notion of $d$-$σ$-stability introduced for a nonempty subset of a random metric space can be regarded as a special case of the notion of $σ$-stability introduced for a nonempty subset of 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 $L^{\infty}$-module is an $L^{p}$-normed $L^{\infty}$-module iff it is generated by a complete random normed module, from which it is easily seen that the gluing property of an $L^{p}$-normed $L^{\infty}$-module can be derived from the $σ$-stability of the generating random normed module, as applications the known and new basic facts of module duals for $L^{p}$-normed $L^{\infty}$-modules can be obtained, in a simple and direct way, from the theory of random conjugate spaces of random normed modules. Third, we prove that a random normed space is order complete iff it is complete with respect to the $(\varepsilon,λ)$-topology, 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 $X\times B$ of a nonempty set $X$ and a complete Boolean algebra $B$ is regular iff it can be induced by a $B$-valued Boolean metric $d$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_08421 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The relations among the notions of various kinds of stability and their applications Guo, Tiexin Mu, Xiaohuan Tu, Qiang Functional Analysis Metric Geometry Primary 18F15, 46A16, 46H25, 53C23 First, we prove that a random metric space can be isometrically embedded into a complete random normed module, as an application of which, it is easy to see that the notion of $d$-$σ$-stability introduced for a nonempty subset of a random metric space can be regarded as a special case of the notion of $σ$-stability introduced for a nonempty subset of 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 $L^{\infty}$-module is an $L^{p}$-normed $L^{\infty}$-module iff it is generated by a complete random normed module, from which it is easily seen that the gluing property of an $L^{p}$-normed $L^{\infty}$-module can be derived from the $σ$-stability of the generating random normed module, as applications the known and new basic facts of module duals for $L^{p}$-normed $L^{\infty}$-modules can be obtained, in a simple and direct way, from the theory of random conjugate spaces of random normed modules. Third, we prove that a random normed space is order complete iff it is complete with respect to the $(\varepsilon,λ)$-topology, 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 $X\times B$ of a nonempty set $X$ and a complete Boolean algebra $B$ is regular iff it can be induced by a $B$-valued Boolean metric $d$ 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. |
| title | The relations among the notions of various kinds of stability and their applications |
| topic | Functional Analysis Metric Geometry Primary 18F15, 46A16, 46H25, 53C23 |
| url | https://arxiv.org/abs/2401.08421 |