泛函分析中赋范线性空间上几种收敛之研究.doc

第 第 PAGE \* MERGEFORMAT 2 页 摘要：本文主要研究了泛函分析中赋范线性空间上几种收敛之间的关系.在书本的基础上对序列的强收敛与弱收敛之间、算子列的一致收敛、强收敛与弱收敛的关系进行进一步地探究与讨论．本文给出了各种收敛性的定义，并讨论了各收敛性之间的关系，给出相应命题，并加以证明.在研究点列收敛性的关系时，本文在Schauder基存在的假设下，给出了Banach空间中强收敛与弱收敛的等价关系的一个充分条件.在研究算子列收敛性的关系时，本文对赋范线性空间加以较强的限制，得到了赋范线性空间中一致收敛与强收敛的等价性和强收敛与弱收敛的等价性. 关键字：赋范线性空间 , 强收敛 , 弱收敛 , 一致收敛 , Schauder基 ABSTRACT:In this paper, we study the relationships of various convergence in normed linear space in Functional Analysis. We will discuss the relationships between strong and weak convergence of sequences, as well as the relationships of uniform convergence, strong convergence and weak convergence of operator sequences. More precisely, we will give the definitions of convergences, discuss the relationships of the various convergences, and prove some results that explain the relationships of the convergence. Moreover, under the assumptions of Schauder basis，we will give the equivalence relation between strong and weak convergence . When we study the convergence of the operator sequences , we try to give strong restrictions on normed linear space, so that we gain the equivalence of the uniform and the strong convergence as well as the equivalence of the strong and weak convergence. KEY WORDS:Normed linear space; Strong Convergence; Weak Convergence; Uniform Convergence；Schauder basis 目 录 引言……………………………………………………………………………………………3 1符号说明和预备知识…………………………………………………………………3 1.1 符号说明………………………………………………………………………………3 1.2 相关定义………………………………………………………………………………3 1.3 重要引理………………………………………………………………………………4 2主要结果…………………………………………………………………………………5 2.1 点列的强收敛与弱收敛……………………………………………………………5 2.1.1强收敛一定弱收敛，弱收敛不一定强收敛……………………………5 2.1.2 强收敛与弱收敛等价的一个充分条件…………………………………5 2.2 算子列的强收敛、弱收敛与一致收敛…………………………………………6 2.2.1 一致收敛一定强收敛，强收敛不一定一致收敛 ……………………6 2.2.2 强收敛一定弱收敛，弱收敛不一定强收敛…………………………7 2.2.3 较强限制条件下的一致收敛、强收敛、弱收敛的关系…………8 3小结 ………………………………………………………………………………………9 4谢辞 ………………………………………………………………………………………9 5参考文献 ………………………………………………………………………………10 6附录： 外文翻译原文…………