Isosceles正交注记

A note on Isosceles orthogonal

在实赋范线性空间E(dimE ≥ 2 )中证明 :当E中向量x ,y线性无关 ,且‖x‖ ≥‖y‖ >0时 ,存在唯一的a ∈R使得x+‖y‖ (y +ax)‖y +ax‖ =x- ‖y‖ (y +ax)‖y +ax‖即在x与y生成的平面上xIsosceles正交且只正交于一个范数是‖y‖的向量 .

LetE be a real normed linear space,and the dimension of which be higher than one,and assume thatx,y∈E andx is linearly independent withy.we prove if‖x‖≥‖y‖>0,then there exist a unique real numbera∈R that x+‖y‖(y+ax)‖y+ax‖=x-‖y‖(y+ax)‖y+ax‖ i.e. on the plane formed byx andy,x is Isosceles orthogonal to only one vector whose norm is ‖y‖.