The mapping is defined by , where is an n-dimensional real Euclidean vector space.

In a Euclidean vector space, the reflection in the point situated at the origin is the same as vector negation.

It is easy to extend the Riemann integral to functions with values in the Euclidean vector space R for any n.

Non-linear dimensionality reduction algorithms attempt to map high-dimensional data onto a low-dimensional Euclidean vector space.

In a geometric algebra generated by a Euclidean vector space of dimension 2 or 3, all sums of 2-blades may be written as a single 2-blade.

A rotation group is a group of orientation-preserving orthogonal transformations of a Euclidean vector space, which have a common fixed point.

A symplectic form behaves quite differently from a symmetric form, for example, the scalar product on Euclidean vector spaces.

A Euclidean vector space with the group operation of vector addition is an example of a non-compact Lie group.

First, it is shown how Euclidean spinors may be interpreted as entities in the geometric algebra of a Euclidean vector space.

Let V be a finite-dimensional Euclidean vector space, with the standard Euclidean inner product denoted by .

MVU creates a mapping from the high dimensional input vectors to some low dimensional Euclidean vector space in the following steps:

Euclidean vector, a geometric entity endowed with magnitude and direction as well as a positive-definite inner product; an element of a Euclidean vector space.

For a Euclidean vector space it is written or Cℓ(ℝ), where n is the dimension of the vector space ℝ.

This model can be generalized to model an n+1 dimensional hyperbolic space by replacing the real number x by a vector in an n dimensional Euclidean vector space.