# Hyperplane

From Encyclopedia of Mathematics

*in a vector space over a field *

The image (under a translation) of a vector subspace with one-dimensional quotient space , i.e. a set of the form for a certain . If , the hyperplane is sometimes called homogeneous. A subset is a hyperplane if and only if

(*) |

for and a certain non-zero linear functional . Here, and are defined by up to a common factor .

In a topological vector space any hyperplane is either closed or is everywhere dense; as defined by formula is closed if and only if the functional is continuous.

**How to Cite This Entry:**

Hyperplane.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Hyperplane&oldid=17985

This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article