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
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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=38653
Hyperplane. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hyperplane&oldid=38653
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article