| 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⊥Jy 4) x⊥Jy 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 the James′,Lp-,isosceles, Pythagorean and (α,β)-orthogonalities defined in E implies Roberts′ orthogonality then E is an abstract Euclidean space. |
