Definition:Affine Space/Weyl's Axioms

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Definition

Let $K$ be a field.

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

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

$- : \mathcal E \times \mathcal E \to V$

satisfying the following associativity conditions:

\((W1)\)   $:$     \(\displaystyle \forall p \in \mathcal E: \forall v \in V: \exists ! q \in \mathcal E:\) \(\displaystyle v = q - p \)             
\((W2)\)   $:$     \(\displaystyle \forall p, q, r \in \mathcal E:\) \(\displaystyle \paren{r - q} +_V \paren{q - p} = r - p \)             


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


Also see


Source of Name

This entry was named for Leonhard Paul Euler.


Sources