关于向量与矩阵对偶范数的注记  

Remarks on the dual norm of vector and matrix

作　　者：任芳国[1] 和嘉琪 REN Fang-guo;HE Jia-qi(School of Mathematics and Statistics, Shaanxi Normal University, Xi'an 710119, Shaanxi, China)

机构地区：[1]陕西师范大学数学与统计学院,陕西西安710119

出　　处：《宝鸡文理学院学报（自然科学版）》2022年第2期1-7,22,共8页Journal of Baoji University of Arts and Sciences(Natural Science Edition)

基　　金：国家自然科学基金项目(11471200)。

摘　　要：目的深入刻画线性空间C^(n)与M_(n)中常见的重要的范数的对偶范数。方法利用对偶范数定义及范数的特性,通过Holder不等式、对偶原理、排序不等式、奇异值的Weyl不等式及Neumann不等式进行研究。结果给出C^(n)上l_(p)-范数与k-范数及M_(n)上Schatten p-范数和Ky Fan k-范数的表示,并给出M_(n)上算子范数的特性。结论完善了线性空间C^(n)与M_(n)中对偶范数的性质,为利用范数解决数值计算问题奠定了理论基础。Purposes—To characterize the dual norm of important norm in inear space C^(n)and M_(n).Methods—The aforesaid purposes are achieved by using the definition of dual norm and the characteristics of norm,Holder inequality,duality principle,sequence inequality,Weyl inequality of singular value and Neumann inequality.Results—Given are the expressions of l_(p)-norm and k-norm on C^(n)and Schatten p-norm and Ky Fan k-norm on M_(n),and so are the characteristics of operator norm on M_(n).Conclusions—The properties of dual norms in linear space C^(n)and M_(n)are improved,which lays a foundation for using norms to solve numerical problems.

关 键 词：向量范数 矩阵范数 对偶范数 算子范数 酉不变范数 

分 类 号：O151.21[理学—数学]