Euclidean space

1. What is meant by Euclidean space?
2. What is the difference between Euclidean and Cartesian space?
3. What is Euclidean space in real analysis?
4. What is the difference between Euclidean space and vector space?

What is meant by Euclidean space?

Euclidean space, In geometry, a two- or three-dimensional space in which the axioms and postulates of Euclidean geometry apply; also, a space in any finite number of dimensions, in which points are designated by coordinates (one for each dimension) and the distance between two points is given by a distance formula.

What is the difference between Euclidean and Cartesian space?

A Euclidean space is geometric space satisfying Euclid's axioms. A Cartesian space is the set of all ordered pairs of real numbers e.g. a Euclidean space with rectangular coordinates.

What is Euclidean space in real analysis?

Technical definition

A Euclidean vector space is a finite-dimensional inner product space over the real numbers. A Euclidean space is an affine space over the reals such that the associated vector space is a Euclidean vector space. ... The dimension of a Euclidean space is the dimension of its associated vector space.

What is the difference between Euclidean space and vector space?

A vector space is a structure composed of vectors and has no magnitude or dimension, whereas Euclidean space can be of any dimension and is based on coordinates.