Definition:Affine Space

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

Denote with $\mathop{\tilde +}$ the operation of vector addition on $V$.

An affine space $\mathcal E$ 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:

Notation
Almost invariably the same symbol (usually $+$) is used for the addition $\mathop{\tilde +}: V \times V \to V$ in the vector space and the addition $+: \mathcal E \times V \to \mathcal E$ in the affine space.

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 $\vec{p q} = q - p$.