Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in
Ancient … See more

Euclidean space was introduced by ancient Greeks as an abstraction of our physical space. Their great innovation, appearing in Euclid's Elements was to build and prove all geometry by starting from a few very basic properties, … 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

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