关于随机赋范空间与随机内积空间的某些基本理论(英文)

首先提出随机度量空间定义的另一个提法，这提法不仅等价于原始的定义而且也使随机度量空间自动归入广义度量空间的框架，也考虑了关于拓扑结构的某些新的问题；循着同样的思路，对随机赋范空间的定义也作了新的处理并同时简化了随机赋范模的定义．其次本文也证明了一个Ｅ－范空间的商空间等距同构于一个典型的Ｅ－范空间；进一步，在概率赋范空间的框架下证明了一个概率赋伪范空间是伪内积生成空间的充要条件是它等距同构于一个Ｅ－内积空间，这回答了Ｃ．Ａｌｓｉｎａ与Ｂ．Ｓｃｈｗｅｉｚｅｒ等人新近提出的公开问题．最后，本文转向了它的中心部分──关于随机内积空间的研究，对随机内积空间中的特有且复杂的正交性作较系统的讨论，论证了只有几乎处处正交性才是唯一合理的正交性概念，在此基础上本文尤其将Ｇ．Ｓｔａｍｐａｃｃｈｉａ的在众多学科中都有多种用途的一般投影定理（或称变分不等式解存在性定理）以适当形式推广到完备实随机内积模上．

Some Basic Theories of Random Normed Linear Spaces and Random Inner Product Spaces

This paper begins by giving another formulation of the original definition of a random metric space. The present formulation is not only equivalent to the original one but also makes a random metric space automatically fall into the framework of a generalized metric space, and some new problems on topo- logical structures are also considered. Motivated by the formulation of a random metric space mentioned above, this paper, then, explicitly presents a corresponding form of the definition of a random normed space and simplifies the definition of a random normed module. Meanwhile this Paper also shows that the quotient space of an E-norm space is isomorphically isometric to a canonical E-norm space, further under the framework of probabilistic pseudonormed spaces, this paper shows a probabilistic pseudonormed linear space is a pseudo-inner product generated space iff it is isomorphically isometric to an E-inner product space (this result answers an open problem recently presented by C. Alsina, B. Schweizer, et al. ). Finally, based on the preceding preliminaries, this paper turns to its central part: the investigations on basic theo- ries of random inner product spaces and random inner product modules: in this part, this paper gives a deep discussion of interesting and complicated orthogonality problems which shows that compared with the other weaker orthogonality, only this kind of orthogonalitiy called almost sure orthogonality possesses almost all the well-conditioned properties which are comparable to those the orthogonality in ordinary inner product spaces has, this also further motivates us to generalize G. Stampacchia's general projection theorem from real Hilbert spaces, in an appropriate form, to real complete random inner product modules.