报告题目 (Title)：The relations among the notions of various kinds of stability and their applications

中文标题：各种稳定性概念之间的关系及其应用

报告人 (Speaker)：郭铁信（中南大学）

报告时间 (Time)：2024年4月3日(周三) 10:00

报告地点 (Place)：校本部GJ303

邀请人(Inviter)：席东盟、李晋、张德凯、吴加勇

主办部门：理学院数学系

报告摘要：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_δ-stability introduced for a nonempty subset of a random metric space can be regarded as a special case of the notion of δ-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_δ-stable random metric space to be stably compact. Second, we prove that an L^∞-module is an L^p-normed L^∞-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^∞-module can be derived from the δ-stability of the generating random normed module, as applications the known and new basic facts of module duals for L^p-normed L^∞-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 (ε,λ)-topology, as an application it is proved that the d-decomposability of an order complete random normed space is exactly its d-δ-stability. Finally, we prove that an equivalence relation on the product space X×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.