随机局部凸模中的Krein-Milman定理及其应用  

The Krein-Milman theorem in random locally convex modules and its applications

作　　者：郭铁信[1] 王亚超 唐艳 Tiexin Guo;Yachao Wang;Yan Tang

机构地区：[1]中南大学数学与统计学院,长沙410083 [2]四川大学数学学院,成都610064 [3]重庆工商大学数学与统计学院,重庆400067

出　　处：《中国科学：数学》2023年第12期1667-1684,共18页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11971483)资助项目。

摘　　要：本文首先深入研究随机局部凸模中的稳定紧集,证明它关于(ε,λ)-拓扑Tε,λ是完备的,并给出它的一个简明的特征,即一个σ-稳定集是稳定紧的当且仅当它的每个具有有限交性质的由σ-稳定的Tε,λ-闭子集组成的σ-稳定族必有非空交.在此基础上,对定义在稳定紧集上的σ-稳定的、真的、下半连续的L^(0)-值函数给出相应的Weierstrass定理,并由此证明一个稳定紧的L^(0)-凸集必为L^(0)-凸紧的.然后,对L^(0)-凸集引进L^(0)-端点的概念并对L^(0)-凸紧集证明相应的Krein-Milman定理,同时给出这个推广的Krein-Milman定理与经典的Krein-Milman定理的某些有趣的比较与联系.最后,作为应用,证明定义在L^(0)-凸紧集上的真下半连续L^(0)-拟凸函数f必达到最小值.进一步地,如果f还是L^(0)-仿射的,那么f的最小值也可以在L^(0)-端点达到.This paper first deeply studies stably compact sets of a random locally convex module by proving that they are complete with respect to the(ε,λ)-topology Tε,λand characterizing them in the way that each-stable family of-stable Tε,λ-closed subsets of them with the finite intersection property has a nonempty intersection.Based on the preliminaries,we further give the Weierstrass theorem for a proper lower semicontinuous L^(0)-valued σ-stable function defined on a stably compact set,which implies that a stably compact L^(0)-convex set must be L^(0)-convexly compact.Then,we introduce the notion of an L^(0)-extreme point for an L^(0)-convex set and prove the corresponding Krein-Milman theorem for an L^(0)-convexly compact set.Besides,some interesting connection and comparison between the generalized Krein-Milman theorem and the classical one for a compact convex set in a locally convex space are also given.Finally,as an application of the generalized Krein-Milman theorem,we prove that a proper,lower semicontinuous,σ-stable and L^(0)-quasiconvex function on an L^(0)-convexly compact set of a random locally convex module can attain its minimum value,and further if in addition,f is L^(0)(F)-valued and L^(0)-affine,then f can attain its minimum value at some L^(0)-extreme point of the L^(0)-convexly compact set.

关 键 词：随机局部凸模 稳定紧集 L^(0)-凸紧集 L^(0)-端点 Krein-Milman定理 

分 类 号：O177[理学—数学]