Euclidean space

Definition

Euclidean $n$ -space is a metric space $(E,d)$ with the property that the group of isometries is transitive and is isomorphic to an $n$ -dimensional Euclidean vector space. To be more precise, we are saying that there exists an $n$ -dimensional Euclidean vector space $V$ with inner product $\langle \cdot,\cdot\rangle$ and a mapping$$ +: E\times V\to E$$ such that the following hold:

1. For all $x,y\in E$ there exists a unique $u\in V$ satisfying$$ y=x+u,\quad d(x,y)^2=\langle u,u\rangle$$
2. For all $x,y\in E$ and all $u\in V$ we have$$ d(x+u,y+u)=d(x,y)$$
3. For all $x\in E$ and all $u,v\in V$ we have$$ (x+u)+v=x+(u+v)$$

Putting it more succinctly: $V$ acts transitively and effectively on $E$ by isometries.

Remarks.

• The difference between Euclidean space and a Euclidean vector space is one of loss of structure. Euclidean space is a Euclidean vector space that has ``forgotten'' its origin.
• A 2-dimensional Euclidean space is often called a Euclidean plane.