Definition:Euclidean Space

From ProofWiki
Jump to navigation Jump to search


This page has been identified as a candidate for refactoring of medium complexity.
In particular: 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 $\mathsf{Pr} \infty \mathsf{fWiki}$.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Refactor}} from the code.



It has been suggested that this page or section be merged into Definition:Euclidean Metric on Real Vector Space.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Mergeto}} from the code.



It has been suggested that this page be renamed.
In particular: The term "Eucldean space" is too wide. Needs to be "Cartesian space under Euclidean metric" -- or better, just make Definition:Euclidean Metric the main page.
To discuss this page in more detail, feel free to use the talk page.


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 (real) 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 $\struct {\Q^n, d}$ is a (rational) Euclidean $n$-space.


Complex Euclidean Space

Let $\C$ be the complex plane.

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

Then $\struct {\C, d}$ is a (complex) 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 $M = \struct {S^n, d}$ 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, 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

Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Euclidean_Space&oldid=644586"