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§2.欧几里得空間 8.欧几里得空間的定义,前面我們已經看到,在一个線性空間中可以定义平面、直線、平行和相交的概念等,不过决不能以線性空間的术語定义像向量的長、向量之間的角这样的基本几何概念,我們將根据用公理法定义的数量乘积的概念来引入这些概念*)。定义7.如果線性空間R的每一对向量x,y与某一实数(x,y)对应,而且这一对应具有下面的性質:
§ 2. Euclidean space 8. The definition of Euclidean space. We have already seen in the foregoing that the concept of planes, lines, parallels, and intersections can be defined in a linear space, but it cannot be done in a linear space. The language defines the basic geometric concepts such as the length of the vector, the angle between the vectors, and we will introduce these concepts based on the concept of quantitative product defined by the axiom method. Definition 7. If each pair of vectors x, y of a linear space R corresponds to a real number (x, y), and this correspondence has the following properties: