# Definition:Affine Space

## Definition

### Associativity Axioms

Let $K$ be a field.

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

Let $\mathcal E$ be 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:

 $(A1)$ $:$ $\displaystyle \forall p, q \in \mathcal E:$ $\displaystyle p + \left({q - p}\right) = q$ $(A2)$ $:$ $\displaystyle \forall p \in \mathcal E: \forall u, v \in V:$ $\displaystyle \left({p + u}\right) + v = p + \left({u +_V v}\right)$ $(A3)$ $:$ $\displaystyle \forall p, q \in \mathcal E: \forall u \in V:$ $\displaystyle \left({p - q}\right) +_V u = \left({p + u}\right) - q$

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

### Group Action

Let $K$ be a field.

Let $\left({V, +_V, \circ}\right)$ be a vector space over $K$.

Let $\mathcal E$ be a set.

Let $\phi: \mathcal E \times V \to \mathcal E$ be a free and transitive group action of $\struct{V, +_V}$ on $\mathcal E$.

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

### Weyl's Axioms

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.

Let $\tuple{\mathcal E, +, -}$ be an affine space.

Then the mapping $+$ is called affine addition.

### Subtraction

Let $\tuple{\mathcal E, +, -}$ be an affine space.

Then the mapping $-$ is called affine subtraction.

### Tangent Space

Let $\tuple {\mathcal E, +, -}$ be an affine space.

Let $V$ be the vector space that is the codomain of $-$.

Then $V$ is the tangent space of $\mathcal E$.

### Vector

Let $\mathcal E$ be an affine space.

Let $V$ be the tangent space of $\mathcal E$.

Any element $v$ of $V$ is called a vector.

### Point

Let $\mathcal E$ be an affine space.

Any element $p$ of $\mathcal E$ is called a point.