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

An isometry between two metric spaces is a bijection preserving the distance, that is
$${\displaystyle d(f(x),f(y))=d(x,y).}$$
In the case of a Euclidean vector space, an isometry that … See more

The definition of Euclidean spaces that has been described in this article differs fundamentally of Euclid's one. In reality, Euclid did not define … 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 … See more

The vector space $${\displaystyle {\overrightarrow {E}}}$$ associated to a Euclidean space E is an inner product space. This implies a symmetric bilinear form 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 $${\displaystyle \mathbb {R} ^{n},}$$ this topology is also the product topology.
The See more