Euclidean space

In mathematics, a Euclidean space is a vector space of dimension n over the field of real numbers, where n is a finite natural number not equal to zero. It is isomorphic to the space $\mathbb {R} ^{n}$ of ordered n-tuples ("columns") of real numbers and hence is usually identified with the latter. In addition, a distance d(x,y) must be defined between any two elements x and y of a Euclidean space, i.e., a Euclidean space is a metric space.

The majority of texts require the distance to be defined by means of a positive definite inner product,

$d(\mathbf {x} ,\mathbf {y} )\equiv \langle \mathbf {x} -\mathbf {y} ,\mathbf {x} -\mathbf {y} \rangle \equiv \left[\sum _{i=1}^{n}(x_{i}-y_{i})^{2}\right]^{\frac {1}{2}},$

where x_i are the components of x and y_i of y. Thus, most commonly a Euclidean space is defined as the real inner product space $\mathbb {R} ^{n}$.

However, this definition does not completely agree with the space appearing in the geometry of Euclid. After all, it was almost 2000 years after Euclid's Elements when Descartes introduced ordered 2-tuples and 3-tuples to describe points in the plane and the 3-dimensional space.

One can introduce the following affine map on $\mathbb {R} ^{n}$:

$\mathbf {x} \mapsto \mathbf {x} '=\mathbf {A} \mathbf {x} +\mathbf {c} ,\quad \mathbf {x} ,\mathbf {x} '\in \mathbb {R} ^{n},$

where A is a real n×n matrix and c is an ordered n-tuple of real numbers. If A is an orthogonal matrix this map leaves distances invariant and is called an affine motion; if furthermore c = 0 it is a rotation. If A = E (the identity matrix), it is a translation. In the classical Euclidean geometry it is irrelevant at which points in space the geometrical objects (circles, triangles, Platonic solids, etc.) are located. This means that Euclid assumed implicitly the invariance of his geometry under the set of affine motions.

A real inner product space equipped with an affine map is an affine space. Thus, formally, the space of high-school geometry is the 2- or 3-dimensional affine space equipped with inner product and, in summary, a Euclidean space may be defined as an n-dimensional affine space with inner product.