Difference between revisions of "Euclidean space"
From Encyclopedia of Mathematics
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| − | + | A space the properties of which are described by the axioms of [[Euclidean geometry]]. In a more general sense, a Euclidean space is a finite-dimensional real [[vector space]] $\mathbb{R}^n$ with an [[inner product]] $(x,y)$, $x,y\in\mathbb{R}^n$, which in a suitably chosen ([[Cartesian orthogonal coordinate system|Cartesian]]) coordinate system $x=(x_1,\ldots,x_n)$ and $y=(y_1,\dots,y_n)$ is given by the formula \begin{equation} (x,y)=\sum_{i=1}^{n}x_i y_i. \end{equation} | |
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| + | Sometimes the phrase "Euclidean space" stands for the case $n=3$, as opposed to the case $n=2$ "Euclidean plane", see [[#References|[1]]], Chapts. 8, 9. | ||
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| + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> M. Berger, "Geometry" , '''I''' , Springer (1987). {{ZBL|0606.51001}} {{DOI|10.1007/978-3-540-93815-6}}</TD></TR></table> | ||
Latest revision as of 14:38, 1 November 2023
A space the properties of which are described by the axioms of Euclidean geometry. In a more general sense, a Euclidean space is a finite-dimensional real vector space $\mathbb{R}^n$ with an inner product $(x,y)$, $x,y\in\mathbb{R}^n$, which in a suitably chosen (Cartesian) coordinate system $x=(x_1,\ldots,x_n)$ and $y=(y_1,\dots,y_n)$ is given by the formula \begin{equation} (x,y)=\sum_{i=1}^{n}x_i y_i. \end{equation}
Comments
Sometimes the phrase "Euclidean space" stands for the case $n=3$, as opposed to the case $n=2$ "Euclidean plane", see [1], Chapts. 8, 9.
References
| [1] | M. Berger, "Geometry" , I , Springer (1987). Zbl 0606.51001 DOI 10.1007/978-3-540-93815-6 |
How to Cite This Entry:
Euclidean space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euclidean_space&oldid=13577
Euclidean space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euclidean_space&oldid=13577
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article