Definition:Euclidean Space
in the process of being refactored, or:
has been identified as a candidate for refactoring. In particular:
Split the various versions of this into their own pages, and transclude. But probably not the "Euclidean Topology" section, as we have a precedent that the space and the topology giving rise to it have been defined on just one page.
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Definition
Real Vector Space
Let $\R^n$ be an $n$-dimensional real vector space.
Let $M = \left({\R^n, d}\right)$ where $\displaystyle d \left({x, y}\right) = \left({\sum_{i=1}^n \left({x_i - y_i}\right)^2}\right)^{1 / 2}$.
Then $M = \left({\R^n, d}\right)$ is a metric space.
Such a space is called a Euclidean $n$-space.
The metric $d$ is called the Euclidean Metric.
Rational Euclidean Space
Let $\Q^n$ be an $n$-dimensional vector space of rational numbers.
From Rational Numbers form Metric Space it follows from the above definition it follows that $\Q^n$ is also a Euclidean $n$-space.
Complex Euclidean Space
The set of complex numbers $\C$ is also a metric space, as is proved here.
Euclidean Topology
The topology induced by the Euclidean metric on a Euclidean space $M$ is called the Euclidean topology.
The Euclidean topology is sometimes called the usual topology.
Also see
- Results about Euclidean spaces can be found here.
Source of Name
This entry was named for Euclid.
Bear in mind that Euclid himself did not in fact conceive of the Euclidean space as defined here. It is called that because the geometric space which it gives rise to is Euclidean in the sense that it is consistent with Euclid's fifth postulate.
Sources
- William H. McCrea: Analytical Geometry of Three Dimensions (1942): $\S 1$
- Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{II}: \ 28$
- W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Example $2.2.1$
