概率赋范空间与概率赋准范空间的转换关系

Transformation Between Probabilistic Normed Space and Probabilistic Quasi-normed Space

研究了概率赋范空间和概率赋准范空间的相互转换关系,得到了相互转换的一个充要条件.概率赋范空间不一定是拓扑线性空间,从而不一定是概率赋准范空间.对于概率赋准范空间,举例说明了存在概率赋准范空间,不能找到一个三角函数使之转换成概率赋范空间.最后对α-简单空间的定义进行了推广,讨论了α-简单空间与概率赋范空间的相互转换关系.

This paper studies the transformation relation between probabilistic normed space and probabilistic quasi-normed space and obtains a sufficient and necessary condition for this conversion. A probabilistic normed space is not necessarily a topological linear space and, therefore, may not be a probabilistic qua si-normed space. Examples are given to show that there exists a probabilistic quasi-normed space as well, for which one cannot find a suitable triangle function to transform it to a probabilistic normed space. Finally, the definition of α-simple space is generalized, and the transformation between probabilistic normed space and α-simple space is discussed.