随机度量理论及其应用在我国最近进展的综述

本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度量空间理论和一简单的介绍；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（本称之为F－随机度量理论）的整体关系：主要结果是在随机元生成空间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；主要思想是将随机泛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（本节由作的工作构成）；在第四节，基本作最近提出的随机度量理论的一个新的版本（本称之为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（本称之为E－随机共轭空间理论），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（本节内容整体上由作最近后篇论构成，也尤其提到朱林户等人的重要工作）；在本节我们也相当的篇幅论述F－随机共轭空间理论与E－随机共轭空间理论的内存关系与本质差异。在下紧跟的两节，致力于E－随机共轭空间理论深层次的结果，尤其突出了E－随机赋范模与传统的赋范空间、E－随机共轭空间与经典共轭空间之间的内存联系；在第五节给出了几类E－随机赋范模的E－随机共轭空间的表示定理（主要由作的工作，作与游兆永及林熙合作的工作，还有巩馥州与刘清荣合作的工作组成）；在第六节给出完备E－随机赋范模为随机自反的特征化定理（主要由作及合作的工作组成）；在第六节给出完备E－随机赋范模为随机自反的特征化定理（主要由作及合作的工作组成）。尤其在第五及第六节中，我们给出随机度量理论在随机泛函分析及经典Banach空间中若干实质性的应用；第七节简要给出E－随机赋半范模及E－随机对偶系理论初步；第八节简单阐明随机度量理论与泛函分析的关系；第九节阐明了随机度量理论与概率度量空间理论的关系。最后在第十节结合随机度量理论，Banach空间理论及随机泛函分析对发展随机泛函分析的空间随机化途径的合理性与优越性作了进一步的分析。

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

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 Frandom 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 a new version of random metric theory-E-random metric theory recently 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 Fuzhou) ; 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 8 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.