# Definition:Euclidean Metric/Real Vector Space

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## Definition

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

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

- $\ds \map {d_2} {x, y} := \paren {\sum_{i \mathop = 1}^n \paren {x_i - y_i}^2}^{1 / 2}$

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

### 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:

- $\ds \map {d_2} {x, y} := \sqrt {\paren {x_1 - y_1}^2 + \paren {x_2 - y_2}^2}$

where $x = \tuple {x_1, x_2}, y = \tuple {y_1, y_2} \in \R^2$.

### Complex Plane

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

- $\forall z_1, z_2 \in \C: \map d {z_1, z_2} := \size {z_1 - z_2}$

where $\size {z_1 - z_2}$ denotes the modulus of $z_1 - z_2$.

### Ordinary Space

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

- $\map {d_2} {x, y} := \sqrt {\paren {x_1 - y_1}^2 + \paren {x_2 - y_2}^2 + \paren {x_3 - y_3}^2}$

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

## 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, Euclidean topology 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

- 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{III}$: Metric Spaces: Pythagoras' Theorem - 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 2$: Metric Spaces: Theorem $2.5$ - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.2$: Examples: Example $2.2.1$