---
tag: 0XPR1E85W0
kind: object
name: Euclidean space
aliases: [euclidean-n-space]
msc: [51M05]
instance_of: [09XNAPTJQ7]
sources:
  - url: https://en.wikipedia.org/wiki/Euclidean_space
    locator: Prototypical examples
same_as:
  wikidata: Q17295
  wikipedia: Euclidean space
---

## Description

Let $n$ be a positive integer.
The Euclidean $n$-space is the set $\mathbb{R}^n$ of the $n$-tuples $x = (x_1, \dots, x_n)$ of real numbers, with the dot product
$$x \cdot y = x_1 y_1 + x_2 y_2 + \dots + x_n y_n.$$
The elements of $\mathbb{R}^n$ are its points, and also its vectors.
The dot product gives the length $\|x\| = \sqrt{x \cdot x}$ of a vector $x$.
It also gives the distance $\|x - y\|$ between two points $x$ and $y$, and the angle between two nonzero vectors.
For $n = 2$ this is the [[Euclidean plane|0XW7V9ZZ05]], and for $n = 3$ it is the space of elementary solid geometry.

## Remarks

In a more abstract form, a Euclidean space is a real [[affine space|09XNAPTJQ7]] of finite dimension whose space of translations has an inner product. Each such space of dimension $n$ is isometric to $\mathbb{R}^n$ with the dot product.
The notation $\mathbb{E}^n$ is also common for the Euclidean $n$-space.

## Explanations

### L1 human

The Euclidean $n$-space is the space of ordinary geometry in $n$ dimensions. A point is a list of $n$ coordinates. The dot product lets you measure lengths and angles, so the familiar facts of school geometry hold in each dimension.