Euclidean -space is the Space of all -tuples of Real Numbers, (, , ..., ) and is denoted . is a Vector Space and has Lebesgue Covering Dimension . Elements of are called -Vectors. is the set of Real Numbers (i.e., the Real Line), and is called the Euclidean Plane. In Euclidean space, Covariant and Contravariant quantities are equivalent so .
See also Euclidean Plane, Real Line, Vector
Gray, A. ``Euclidean Spaces.'' §1.1 in Modern Differential Geometry of Curves and Surfaces.
Boca Raton, FL: CRC Press, pp. 2-4, 1993.