Definition:Affine Space/Weyl's Axioms

Definition

Let $K$ be a field.

Let $\struct{V, +_V, \circ}$ be a vector space over $K$.

Let $\EE$ be a set on which a mapping is defined:

$- : \EE \times \EE \to V$

satisfying the following associativity conditions:

$(\text W 1)$	$:$			$\ds \forall p \in \EE: \forall v \in V: \exists ! q \in \EE:$	$\ds v = q - p $
$(\text W 2)$	$:$			$\ds \forall p, q, r \in \EE:$	$\ds \paren{r - q} +_V \paren{q - p} = r - p $

Then the ordered pair $\tuple {\EE, -}$ is an affine space.

This entry was named for Hermann Klaus Hugo Weyl.

\((\text W 1)\)	$:$			\(\ds \forall p \in \EE: \forall v \in V: \exists ! q \in \EE:\)	\(\ds v = q - p \)
\((\text W 2)\)	$:$			\(\ds \forall p, q, r \in \EE:\)	\(\ds \paren{r - q} +_V \paren{q - p} = r - p \)