有界集的球包与完备集相关问题的研究
发布时间：2018-07-17 03:45
【摘要】：1911年Meissner在研究欧氏空间的等宽集时引入了完备集的概念,在有界集外增加一点不增加集合的直径,则称该有界集是完备的。关于完备集,很多数学家在一般的实Banach空间特别是有限维实Banach空间中围绕着完备集及其相关性质,集合的完备化映射以及与完备集相关的若干问题做了一系列重要的工作,但是仍有很多问题未被解决。同时,我们知道,有界集的球包与完备集的性质以及完备化集的唯一性有着密切的联系并且在集合的完备化过程中扮演着重要的角色。本文在前人的研究基础上研究集合及其宽球包和紧球包的关系。本文首先回顾了完备集这一概念的来源——等宽集的一些基本性质,完备集的一些性质以及与完备集相关的若干研究问题和相关结论。其次,介绍了几个特殊赋范线性空间的范数以及关于直径点和球的一些相关知识。最后,研究了Banach空间中有界集宽球包和紧球包的边界结构、有界集及其球包的直径点以及有界集到其球包边界的距离。此外,还给出了有界集的宽球包是球的一个充要条件。
[Abstract]:In 1911, Meissner introduced the concept of complete set when he studied the equal-width set of Euclidean space. Adding a single point outside the bounded set without increasing the diameter of the set, the bounded set is said to be complete. On complete sets, many mathematicians have done a series of important work in general real Banach spaces, especially in finite dimensional real Banach spaces, around complete sets and their related properties, complete mapping of sets and some problems related to complete sets. But there are still many unsolved problems. At the same time, we know that the ball packet of bounded set is closely related to the properties of complete set and the uniqueness of complete set and plays an important role in the process of complete set. In this paper, based on the previous studies, we study the relation between the set and its broad and compact spherical packets. In this paper, we first review some basic properties of the origin of the concept of complete set, some properties of complete set, and some research problems and relevant conclusions related to complete set. Secondly, the norm of some special normed linear spaces and some relevant knowledge about diameter points and balls are introduced. Finally, the boundary structure of the bounded set, the diameter points of the bounded set and its ball packet, and the distance between the bounded set and the boundary of the bounded set are studied in Banach space. In addition, a necessary and sufficient condition for the wide ball packet of a bounded set to be a ball is given.
【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】：O177
本文编号：2128859
[Abstract]:In 1911, Meissner introduced the concept of complete set when he studied the equal-width set of Euclidean space. Adding a single point outside the bounded set without increasing the diameter of the set, the bounded set is said to be complete. On complete sets, many mathematicians have done a series of important work in general real Banach spaces, especially in finite dimensional real Banach spaces, around complete sets and their related properties, complete mapping of sets and some problems related to complete sets. But there are still many unsolved problems. At the same time, we know that the ball packet of bounded set is closely related to the properties of complete set and the uniqueness of complete set and plays an important role in the process of complete set. In this paper, based on the previous studies, we study the relation between the set and its broad and compact spherical packets. In this paper, we first review some basic properties of the origin of the concept of complete set, some properties of complete set, and some research problems and relevant conclusions related to complete set. Secondly, the norm of some special normed linear spaces and some relevant knowledge about diameter points and balls are introduced. Finally, the boundary structure of the bounded set, the diameter points of the bounded set and its ball packet, and the distance between the bounded set and the boundary of the bounded set are studied in Banach space. In addition, a necessary and sufficient condition for the wide ball packet of a bounded set to be a ball is given.
【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】：O177
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