# Definition:Affine Space

## Definition

### Associativity Axioms

Let $K$ be a field.

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

Let $\EE$ be a set on which two mappings are defined:

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

satisfying the following associativity conditions:

 $(\text A 1)$ $:$ $\ds \forall p, q \in \EE:$ $\ds p + \paren {q - p} = q$ $(\text A 2)$ $:$ $\ds \forall p \in \EE: \forall u, v \in V:$ $\ds \paren {p + u} + v = p + \paren {u +_V v}$ $(\text A 3)$ $:$ $\ds \forall p, q \in \EE: \forall u \in V:$ $\ds \paren {p - q} +_V u = \paren {p + u} - q$

Then the ordered triple $\struct {\EE, +, -}$ is an affine space.

### Group Action

Let $K$ be a field.

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

Let $\EE$ be a set.

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

Then the ordered pair $\struct {\EE, \phi}$ is an affine space.

### Weyl's Axioms

Let $K$ be a field.

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

Let $\EE$ 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.

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

Then the mapping $+$ is called affine addition.

### Subtraction

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

Then the mapping $-$ is called affine subtraction.

### Tangent Space

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

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

Then $V$ is the tangent space of $\EE$.

### Vector

Let $\EE$ be an affine space.

Let $V$ be the tangent space of $\EE$.

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

### Point

Let $\EE$ be an affine space.

Any element $p$ of $\EE$ is called a point.