# Definition:Euclidean Metric/Real Vector Space

## Contents

## Definition

Let $\R^n$ be an $n$-dimensional real vector space.

The **Euclidean metric** on $\R^n$ is defined as:

- $\displaystyle d_2 \left({x, y}\right) := \left({\sum_{i \mathop = 1}^n \left({x_i - y_i}\right)^2}\right)^{1 / 2}$

where $x = \left({x_1, x_2, \ldots, x_n}\right), y = \left({y_1, y_2, \ldots, y_n}\right) \in \R^n$.

### Real Number Line

On the real number line, the Euclidean metric can be seen to degenerate to:

- $\map d {x, y} := \sqrt {\paren {x - y}^2} = \size {x - y}$

where $\size {x - y}$ denotes the absolute value of $x - y$.

### Real Number Plane

The **Euclidean metric** on $\R^2$ is defined as:

- $\displaystyle d_2 \left({x, y}\right) := \sqrt{\left({x_1 - y_1}\right)^2 + \left({x_2 - y_2}\right)^2}$

where $x = \left({x_1, x_2}\right), y = \left({y_1, y_2}\right) \in \R^2$.

### Complex Plane

The **Euclidean metric** on $\C$ is defined as:

- $\displaystyle \forall z_1, z_2 \in \C: d \left({z_1, z_2}\right) := \left\vert{z_1 - z_2}\right\vert$

where $\left\vert{z_1 - z_2}\right\vert$ denotes the modulus of $z_1 - z_2$.

### Ordinary Space

The **Euclidean metric** on $\R^3$ is defined as:

- $\displaystyle d_2 \left({x, y}\right) := \sqrt{\left({x_1 - y_1}\right)^2 + \left({x_2 - y_2}\right)^2 + \left({x_3 - y_3}\right)^2}$

where $x = \left({x_1, x_2, x_3}\right), y = \left({y_1, y_2, y_3}\right) \in \R^3$.

## Also known as

The **Euclidean metric** is sometimes also referred to as **the usual metric**.

## Also see

- Metric Induces Topology, from which it follows that the Euclidean space is also a topological space.

In this context, the topology induced by the **Euclidean metric** is sometimes called the **usual topology**.

- Results about
**the Euclidean metric**can be found here.

## Source of Name

This entry was named for Euclid.

## Historical Note

Euclid himself did not in fact conceive of the Euclidean metric and its associated Euclidean Space and Euclidean norm.

They bear that name because the geometric space which it gives rise to is **Euclidean** in the sense that it is consistent with Euclid's fifth postulate.

## Sources

- 1962: Bert Mendelson:
*Introduction to Topology*... (previous) ... (next): $\S 2.2$: Metric Spaces: Theorem $2.5$ - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): $\text{III}$: Pythagoras' Theorem - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $2.2$: Examples: Example $2.2.1$