Euclidean space

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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) 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}


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.


[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:
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article