1、欧式空间 Euclidean space

好好斟酌,比较易懂。https://blog.csdn.net/lulu950817/article/details/80424288

https://en.wikipedia.org/wiki/Euclidean_space

These are distances between points and the angles between lines or vectors, which satisfy certain conditions (see below), which makes a set of points a Euclidean space. The natural way to obtain these quantities is by introducing and using the standard inner product (also known as the dot product) on Rn.[3] The inner product of any two real n-vectors x and y is defined by

数学-保持愤怒

where xi and yi are ith coordinates of vectors x and y respectively. The result is always a real number.

设V是实数域R上的线性空间(或称为向量空间),若V上定义着正定对称双线性型g(g称为内积),则V称为(对于g的)内积空间或欧几里德空间(有时仅当V是有限维时,才称为欧几里德空间)。 [3]  具体来说,g是V上的二元实值函数,满足如下关系:
(1)g(x,y)=g(y,x);
(2)g(x+y,z)=g(x,z)+g(y,z);
(3)g(kx,y)=kg(x,y);
(4)g(x,x)>=0,而且g(x,x)=0当且仅当x=0时成立。
这里x,y,z是V中任意向量,k是任意实数
例子:
1. (经典欧几里德空间E^n)在n维实向量空间R^n中定义内积(x,y)=x1y1+...+xnyn,则Rn为欧几里德空间。(事实上,任意一个n维欧几里德空间V等距同构于E^n。)
2. 设V是[0,1]区间上连续实函数全体,则V是R上线性空间,对于如下内积是欧几里德空间:(f,g)定义为fg在[0,1]区间上的积分值。

欧式空间需要满足:

"However, the neural network architectures are not in Euclidean space" 神经网络结构不是欧式空间。