作者:王晓梅，计东海
Authors:WANG Xiaomei,JI Donghai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 = { x₁ ，x₂ ，… ，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 丄_B xi ( Yi≠m) ; ( 2) 对 A 中任意一点 x_m 有 x_i 丄_B x_m ( Yi≠m) 。
Abstract: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 = { x₁ ，x₂ ，… , 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 xm ∈A，there exists x_m 丄_B x_i ( Yi≠m) ; ( 2) for each xm ∈A，there exists x_i 丄_B x_m ( Yi≠m) .


PDF全文下载地址:

可免费Download/下载PDF全文