随机度量理论及其应用在我国最近进展的综述(英文)

Survey of Recent Developments of Random Metric Theory and Its Applications in China (I)

本文旨在全面综述随机度量理论及其应用过去十年在我国发展过程中所获得的主要结果与思想方法．全文由十节组成，第一节对我们工作的背景—概率度量空间与随机质量空间理论作一简单的介绍；第二节给出某些有关随机泛函分析及取值于抽象空间的可测函数的预备知识；第三节阐明随机泛函分析与原始随机度量理论（本文称之为Ｆ－随机质量理论）的整体关系：主要结果是在随机元生成空间上给出自然且合理的随机度量与随机范数的构造，从而将随机元与随机算子理论的研究纳入随机度量理论框架；主要思想是将随机还函分析视为随机度量空间体系上的分析学而统一地发展，从而形成了发展随机泛函分析的一个新的途径—空间随机化途径；除此之外，在本节我们也从随机过程理论的观点出发首次提出对应于随机度量理论原始版本的一种新的随机共轭空间理论（叫作Ｆ－随机共轭空间理论），它的突出优点是能保持象随机过程的样本性质这样更精细的特性（本节由作者的工作构成）；在第四节，基于作者最近提出的随机度量理论的一个新的版本（本文称之为Ｅ－随机度量理论），从传统泛函分析的角度对过去已被发展起来的随机共轭空间理论（本文称之为Ｅ－随机共轭空间理论）的基本结果进行系统整理并给以全新的处理（本节内容整体上?

The central purpose of this paper is to present a complete account of the principal results and ideas currently available in the course of the development of random metric theory and its applications in China. This paper is divided into ten sections. Section 1 is devoted to a brief introduction of the background of our work-the theories of probabilistic metric spaces and random metric spaces ; Section 2 to some preliminaries from random functional analysis and abstract space-valued measurable functions. Section 3 is devoted to a survey of the global relationship between random functional analysis and original random metric theory (called F-random metric theory in this paper): the principal results are smoothly putting random elements, and hence also random operators into the basic frameworks of F-random metric theory by reasonably and naturally constructing F-random metric and F-random norm on the spaces generated by random elements; based on these constructs, we further put together random functional analysis and F- random metric theory to directly lead to a new approach to random functional analysis--the space-randomized approach; besides these, this section also gives, for the first time, the basics of the theory of F-random conjugate spaces(all the results in this section are due to the author). In section 4, based on anew version of random metric theory-E-random metric theory decently presented by the author, we give the basics of the previously developed theory of E-random conjugate spaces (this section mainly consists of the author's work, at the same time Zhu Linhu's important work on random linear functionals on E-norm spaces is in particular mentioned). The following two sections are concerned with the most substantial and deepest parts of the theory of E-random conjugate spaces:Section 5 is devoted to representation theorems of E-random conjugate spaces of several classes of E-random normed modules(this section consists of the auther's work,the joint work of the author with YOU Zhao-yong and LIN Xi,and the joint work of LIU Qing-rong with GONG Fu- zhou); Section 6 to characterizations of E-random reflexive spaces(this section consists of the author's work and the joint work of the author with YOU Zhao-yong and others). Section 7 gives the basics of the theory of E-random seminormed modules together with E-random dualities(this section mainly consists of the author's work). Sections & and 9 are devoted to a brief elucidation of the relations of random metric theory to functional analysis and the theory of probabilistic metric spaces. Section 10 concludes this paper with a further analysis of the space-randomized approach to random functional analysis.