Minkowski空间的Pythagorean正交与内积空间的特征

Pythagorean orthogonality in Minkowski spaces and characterization of inner product spaces

给出Pythagorean正交(简称P正交)齐次元的定义,证明Minkowski平面X为内积空间的充要条件是:X中存在一个非零的Pythagorean正交齐次元。证明任意具有存在性且同时具有齐次性的广义正交蕴含P正交时,Minkowski空间必为内积空间。

The concept of Pythagorean orthogonally homogeneous element is introduced. It is proven that a Minkowski plane X is an inner space if and only if there exists a non-zero Pythagorean orthogonally homogeneous element in X. In addition,it is shown that in a Minkowski space X,if some generalized orthogonality with existence and homogeneity implies Pythagorean orthogonally,then X is an inner product space.