# 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:

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

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

## Source of Name

This entry was named for Hermann Klaus Hugo Weyl.