A Study of Birkhoff Orthogonal Sets in Smooth Banach Spaces - WANG Xiaomei; JI Donghai (Journal of Harbin University of Science and Technology)
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 ，… , xn，xn + 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 xm 丄B xi ( Yi≠m) ; ( 2) for each xm ∈A，there exists xi 丄B xm ( Yi≠m) .