摘要
F.Sullivan提出了L^P-正交的概念,刘证研究了Banach空间中L^P-正交的性质与一致凸空间的关系.本文证明了:设P>1,E是实赋范线性空间,若其内定义的L^p-正交满足:i)齐性;ii)可力性;iii)x(?)_L^py推出X(?)_jy,x,y∈E;iv)x(?)_Jy推出x(?)_L^py,x,y∈E,中任何一项,则E是一个抽象欧氏空间,而且有P=2.另外,本文还为R.C.James的一个未经证实的结论补充了证明.
In this paper we have proved the theorem: Let p>1, and E be a real normed linear space, if the LP-orthogonality in E satisfies one of the following conditions 1) homogeneity 2) additivity 3) x⊥LPy implies x⊥y 4) x⊥y implies x⊥LPy, then E is an abstract Euclidean space and there must be p=2. We also proved anR. C. James' result-If for every element x of a normed linear space E therecan be found a nonzero element orthogonal to x by Roberts' definition in each two dimensional linear subset containing x, then E is an abstract Euclidean space-which had not been proved. Finally, we point out that if one of theJames', LP, isosceles, Pythagorean and (a,β )-orthogonalities defined in E imp-lies Roberts' orthogonality then E is an abstract Euclidean space.
