Euclidean space was introduced by ancient Greeks as an abstraction of our physical space. Their great innovation, … See more

For any vector space, the addition acts freely and transitively on the vector space itself. Thus a Euclidean vector space can be viewed as a Euclidean space that has itself as the associated vector space.
A typical ca… See more

Some basic properties of Euclidean spaces depend only on the fact that a Euclidean space is an affine space. They are called affine properties and include the concepts of lines, subspaces, and parallelism, which are detaile… See more

The vector space associated to a Euclidean space E is an inner product space. This implies a symmetric bilinear form
that is positive definite (that is is always positive for x ≠ 0).
The inner p… See more

An isometry between two metric spaces is a bijection preserving the distance, that is
In the case of a Euclidean vector space, an isometry that maps the origin to the origin preserves the norm
since the n… See more

The Euclidean distance makes a Euclidean space a metric space, and thus a topological space. This topology is called the Euclidean topology. In the case of this topology is also the product topology.
The … See more