 TOPICS # Euclidean Space

Euclidean -space, sometimes called Cartesian space or simply -space, is the space of all n-tuples of real numbers, ( , , ..., ). Such -tuples are sometimes called points, although other nomenclature may be used (see below). The totality of -space is commonly denoted , although older literature uses the symbol (or actually, its non-doublestruck variant ; O'Neill 1966, p. 3). is a vector space and has Lebesgue covering dimension . For this reason, elements of are sometimes 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 .

Euclidean Plane, Point, Pseudo-Euclidean Space, Real Line, Real Number, Tensor, Vector Explore this topic in the MathWorld classroom

Portions of this entry contributed by Christopher Stover

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## References

Gray, A. "Euclidean Spaces." §1.1 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 2-5, 1997.O'Neill, B. Elementary Differential Geometry. San Diego, CA: Academic Press, 1966.

Euclidean Space

## Cite this as:

Stover, Christopher and Weisstein, Eric W. "Euclidean Space." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EuclideanSpace.html