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Euclidean space
1 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 2dimensional Euclidean space is often called a Euclidean plane.
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