Definition:Euclidean Space
We do not really need Euclidean Topology to be a separate subpage and subthread from this, as the "space" and "topology" reall define the same thing.
Until this has been finished, please leave {{refactor}} in the code.
New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.
 It has been suggested that this page or section be merged into Definition:Euclidean Metric/Real Vector Space. (Discuss) |
Contents
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 $\tuple {S^n, d}$ 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 $\struct {\R^n, d}$ 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.
Euclidean Plane
For any real number $a$ let:
- $L_a = \set {\tuple {x, y} \in \R^2: x = a}$
Furthermore, define:
- $L_A = \set {L_a: a \in \R}$
For any two real numbers $m$ and $b$ let:
- $L_{m, b} = \set {\tuple {x, y} \in \R^2: y = m x + b}$
Furthermore, define:
- $L_{M, B} = \set {L_{m, b}: m, b \in \R}$
Finally let:
- $L_E = L_A \cup L_{M, B}$
The abstract geometry $\struct {\R^2, L_E}$ is called the Euclidean plane.
Also see
- 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.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Entry: Euclidean space
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Entry: Euclidean space
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Entry: Euclidean space (Cartesian space)
