Survey of Recent Developments of Random Metric Theory and Its Applications in China (I)
Survey of Recent Developments of Random Metric Theory and Its Applications in China (I)
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发表时间：
2001-06
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通讯作者：
T. Guo
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作者：
T. Guo
本文的主要目的是全面介绍随机度量理论及其应用在中国发展过程中的主要结果和思想。本文共分为十个部分。第一节简要介绍了我们工作的背景--概率度量空间和随机度量空间的理论;第二节介绍了随机泛函分析和抽象空间值可测函数的一些性质。第三节综述了随机泛函分析与原有的随机度量理论（本文称之为F-随机度量理论）的整体关系，主要结果是随机元的光滑配置。通过在随机元生成的空间上合理而自然地构造F随机度量和F随机范数，从而将随机算子引入到F随机度量理论的基本框架中.我们进一步将随机泛函分析与F-随机度量理论结合起来，直接导出了随机泛函分析的一种新方法--空间随机化方法，此外，本节还首次给出了F-随机共轭空间理论的基础（本节的所有结果均为作者所得）。在第4节。基于作者最近提出的一种新的随机度量理论--F随机度量理论，给出了E-随机共轭空间理论的基础（本节主要是作者的工作）。同时特别提到朱林虎关于E-范空间上随机线性泛函的重要工作）。第五节讨论了几类F-随机赋范模的F-随机共轭空间的表示定理（这一节是作者的工作，也是作者与尤兆勇、林曦的合作工作）。第六节F-随机自反空间的刻画（这一节包括作者的工作和作者与尤兆勇等人的合作工作）。第7节给出了E-随机赋范模和F-随机对偶的基本理论（这一节主要是作者的工作）。第8节和第9节致力于简要阐明随机度量理论与泛函分析和概率度量空间理论的关系。第10节结束本文的空间随机化方法随机功能分析的进一步分析。
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 a new version of random metric theory-F 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' a 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 F-random conjugate spaces Section 5 is devoted to representation theorems of F random conjugate spaces of several classes of F-random normed modules (this section consists of the author’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 F-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 F-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.