随机赋范模上非零连续线性泛函的存在性  

Existence of Nonzero Continuous Linear Functionals on Complete Random Normed Modules

作　　者：郭铁信[1] 曾小林[1] 

机构地区：[1]北京航空航天大学理学院数学系,教育部数学、信息与行为重点实验室,北京100083

出　　处：《工程数学学报》2008年第1期117-123,共7页Chinese Journal of Engineering Mathematics

基　　金：国家自然科学基金(10471115)

摘　　要：本文证明了在任意满支承的随机赋范模上存在一个非零连续线性泛函的充要条件是它的基底空间至少存在一个原子;存在足够多非零连续线性泛函的充要条件是它的基底空间本质上由至多可数个原子生成。该结果表明经典的共轭空间理论对随机赋范模是普遍失效的,进一步揭示了随机共轭空间理论对随机赋范模发展的突出重要性。同时本文也包括了许多结果,它们表明许多由随机赋范模生成的经典赋准范空间拥有一个或足够多的非零连续线性泛函的特征成为一目了然!This paper shows that, on a complete random normed module, there exists a nonzero continuous linear functional if and only if there is at least one atom in its base space; and there exist sufficiently many nonzero continuous linear functionals if and only if its base is essentially generated by at most an countable family of atoms. The results show that the theory of classical conjugate spaces is universally invalid for random normed modules, and furthermore, expose the fundamental importance of the theory of random conjugate spaces in the development of random normed modules. At the same time, this paper also contains many results which make them very clear that characterizations for lots of classical quasinormed spaces generated by random normed modules to admit one or sufficiently many nonzero continuous linear functionals.

关 键 词：随机赋范模 连续线性泛函 几乎处处有界随机线性泛函 经典共轭空间 随机共轭空间 

分 类 号：O177.3[理学—数学]