Definition:Affine Space

Definition
Let $V$ be a vector space over any field.

An affine space $\mathcal E$ with difference space $V$ is a set on which two mappings are defined:
 * $+ : \mathcal E \times V \to \mathcal E$
 * $- : \mathcal E \times \mathcal E \to V$

satisfying the following associativity conditions: where $\tilde+$ denotes the sum in the vector space $V$.
 * 1) For all $p,q \in \mathcal E$ we have $p + (q - p) = q$
 * 2) For all $p \in \mathcal E$, $u,v \in V$ we have $(p + u) + v = p + (u\tilde+v)$
 * 3) For all $p,q \in \mathcal E$, $u \in V$ we have $(p - q) \tilde+ u = (p + u) - q$

Notation
Almost invariably the same symbol is used for addition in the vector space and the addition $\mathcal E \times V \to \mathcal E$.

This does not allow any ambiguity as the two mappings have different domains

For elements $p,q \in \mathcal E$, it is common to write $\vec{pq} = q - p$.