---
tag: 0TEJ6QF7K9
kind: definition
name: Affine space
msc: [51N10]
defines: 09XNAPTJQ7
sources:
  - url: https://en.wikipedia.org/wiki/Affine_space
    locator: Definition
---

## Definition

Let $K$ be a field and let $V$ be a vector space over $K$.
An affine space over $V$ is a nonempty set $A$ with a map $A \times V \to A$, written $(p, v) \mapsto p + v$, that has these properties.
For each point $p$ of $A$: $p + 0 = p$.
For each point $p$ of $A$ and all vectors $u$ and $v$ of $V$: $(p + u) + v = p + (u + v)$.
For all points $p$ and $q$ of $A$, there is exactly one vector $v$ of $V$ with $p + v = q$. This vector is written $q - p$ or $\overrightarrow{pq}$.
The elements of $A$ are the points, and $V$ is the space of translations of $A$. The dimension of $A$ is the dimension of $V$.

## Remarks

Each vector space $V$ is an affine space over itself, with the addition of $V$ as the action. In particular $\mathbb{R}^n$ is an affine space over $\mathbb{R}^n$.
For all points $p$, $q$ and $r$, Chasles's relation holds: $(q - p) + (r - q) = r - p$.
Mathlib calls this structure an additive torsor, `AddTorsor`. Its definition works for an additive group $V$ in place of a vector space.
A [[Euclidean space|0XPR1E85W0]] is a real affine space of finite dimension whose space of translations has an inner product.

## Explanations

### L1 human

In an affine space you can subtract two points to get the arrow from one to the other. You can also add an arrow to a point. But you cannot add two points: there is no special origin. A good picture is the plane without its coordinate axes.