光滑Banach空间中Birkhoff正交组的性质  

Study of Birkhoff Orthogonal Sets in Smooth Banach Spaces

作　　者：王晓梅 计东海 WANG Xiaomei;JI Donghai(College of Science,Harbin University of Science and Technology,Harbin 150080,China)

机构地区：[1]哈尔滨理工大学理学院,哈尔滨150080

出　　处：《哈尔滨理工大学学报》2024年第1期143-149,共7页Journal of Harbin University of Science and Technology

基　　金：国家自然科学基金(11571085).

摘　　要：参考内积空间中正交组的定义,在有限维实Banach空间中引入了Birkhoff正交组的概念,并围绕光滑的Banach空间中是否存在所含元素个数超过空间维数的Birkhoff正交组这一问题展开研究。证明了二维光滑的Banach空间中不存在所含元素个数超过空间维数的Birkhoff正交组;三维及以上的光滑Banach空间中不存在所含元素个数超过空间维数且所含元素均为左(右)对称点的Birkhoff正交组。表明了若n(≥3)维光滑的Banach空间中存在Birkhoff正交组A={x_(1),x_(2),…,x_(n),x_(n+1)},则A必不满足以下两个条件:(1)对A中任意一点x_(m)有x_(m)⊥B_(x)_(i)(∀i≠m);(2)对A中任意一点x_(m)有x_(i)⊥_(B)x_(m)(∀i≠m)。Referring to the definition of orthogonal set in inner product space,the concept of Birkhoff orthogonal set is introduced in finite-dimensional real Banach spaces,and the problem of whether there exists a Birkhoff orthogonal set whose number of elements exceeds the space dimension is studied in smooth Banach spaces.It is proved that there is no Birkhoff orthogonal set whose number of elements exceeds the space dimension in two-dimensional smooth Banach spaces.In a smooth Banach space with more than three dimensions,there is no Birkhoff orthogonal set with more elements than the space dimension and all the elements are left(right)symmetric points.It is also proved that if there is a Birkhoff orthogonal set A={x_(1),x_(2),…,x_(n),x_(n+1)}in an n-dimensional(n≥3)smooth Banach space,and then A must not satisfy the following two conditions:(1)for each x_(m)∈A,there exists x_(m)⊥B x_(i)(∀i≠m);(2)for each xm∈A,there exists x_(i)⊥_(B) x_(m)(∀i≠m).

关 键 词：BANACH空间 Birkhoff正交 Birkhoff正交组 光滑性 

分 类 号：O177[理学—数学]