Definition:Euclidean Space

Definition

Let $S$ be one of the standard number fields $\Q$, $\R$, $\C$.

Let $S^n$ be a cartesian space for $n \in \N_{\ge 1}$.

Let $d: S^n \times S^n \to \R$ be the usual (Euclidean) metric on $S^n$.

Then $\left({S^n, d}\right)$ is a Euclidean space.

Special Cases

Real Vector Space

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

Let the Euclidean metric $d$ be applied to $\R^n$.

Then $\left({\R^n, d}\right)$ is a Euclidean $n$-space.

Rational Euclidean Space

Let $\Q^n$ be an $n$-dimensional vector space of rational numbers.

Let the Euclidean Metric $d$ be applied to $\Q^n$.

Then $\left({\Q^n, d}\right)$ is a Euclidean $n$-space.

Complex Euclidean Space

Let $\C$ be the complex plane.

Let $d$ be the Euclidean metric on $\C$.

Then $\left({\C, d}\right)$ is a Euclidean space.

Euclidean Topology

Let $S$ be one of the standard number fields $\Q$, $\R$, $\C$.

Let $S^n$ be a cartesian space for $n \in \N_{\ge 1}$.

Let $M = \left({S^n, d}\right)$ be a Euclidean space.

The topology $\tau_d$ induced by the Euclidean metric $d$ is called the Euclidean topology.

Also see

• Definition:Dot Product
• Definition:Euclidean Norm
• Definition:Euclidean Metric

• Results about Euclidean spaces 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.