拓扑线性空间中的完备集

发布时间：2018-04-18 23:01

  本文选题：宽球包 + 紧球包　； 参考：《哈尔滨理工大学》2017年硕士论文

【摘要】：自从Meissner在1911年首次给出完备集的概念,有关完备集及其特征性质的问题一直受到学者们的广泛关注。完备集及其特征性质的研究不仅在实际应用中具有重要意义,而且在理论研究上具有一定的挑战。本文主要在(实)拓扑线性空间中,关于给定凸体的完备集及其一些性质进行研究。首先,给出了在拓扑线性空间中有界集关于给定凸体的直径和外径的定义。基于有界集关于给定凸体的直径和外径的定义,讨论了在拓扑线性,空间中有界集关于给定凸体的直径和外径的相关结论。其次,基于在实拓扑线性空间中有界集关于给定凸体的宽球包与紧球包、伴随变换和半径函数的定义,讨论有界集关于给定凸体的宽球包与紧球包、伴随变换和半径函数的有关性质,并在此基础上,讨论有界集关于给定凸体完备的充要条件,和有界集关于给定凸体有唯一完备化集的条件。再次,基于J.P.Moreno和R.Schneider提出的(K,u)-完备集的概念,本文重新定义(K,u)-完备集的概念,并讨论在实拓扑线性空间中(K,u)-完备集和唯一完备化集的关系。最后,基于Papini和吴森林给出的在Banach空间中完备化集的构造方法,给出在实拓扑线性空间中有界集关于给定凸体完备化的方法。同时在Eggleston构造完备化集的方法的基础上,在有限维的Banach空间中给出构造完备化集的方法。
[Abstract]:Since the concept of Meissner for the first time in 1911 are complete sets, some properties and characteristics of the complete set of problems has attracted wide attention of scholars. Study on properties of complete sets and features not only have important significance in the practical application, but also has certain challenge in theoretical research. This paper mainly in the (real) linear topological space. Research on complete set for a given convex body and some of its properties. Firstly, gives the definition of boundary in diameter and diameter of a given convex body in topological linear space. Based on bounded sets for a given convex body diameter and diameter of the definition, discusses the related conclusions in topological linear, bounded set diameter and outside for a given convex space. Secondly, based on the real topological linear space bounded sets in a given convex body width and tight bag bag, with the definition of the transformation function and radius, discussion Bounded set for a given convex body width and tight bag bag, with the nature of transformation and function of radius, and on this basis, to discuss the bounded set of sufficient and necessary conditions for a given convex body complete, and bounded set has only one complete set of conditions for a given convex body. Thirdly, based on J.P.Moreno and the R.Schneider (K, U) - the concept of complete set, this paper redefines the concept of (K, U) - complete sets, and discuss in real linear topological space (K, U) - complete set and only complete sets. Finally, methods to construct the complete set in Banach space Papini and Wu Senlin are given based on bounded sets for a given convex completion method in real linear topological space. At the same time in the Eggleston construction method based on the complete set, set of methods is given to construct complete in a finite dimensional Banach space.

【学位授予单位】：哈尔滨理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O189.11

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