Definition:Affine Space/Weyl's Axioms

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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.

Also see

Source of Name

This entry was named for Hermann Klaus Hugo Weyl.