Affine space/Over vector space/Definition

Affine space

An affine space over a ${}K$-vector space ${}V$ is a set ${}E$, together with a mapping

$V\times E\longrightarrow E,(v,P)\longmapsto P+v,$

which satisfies the following three conditions:

1. ${}P+0=P$ for all ${}P\in E$,
2. ${}(P+v)+w=P+(v+w)$ for all ${}v,w\in V$ and ${}P\in E$,
3. For two points ${}P,Q\in E$, there exists exactly one vector ${}v\in V$ such that ${}Q=P+v$.