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| − | ''in a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048490/h0484901.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048490/h0484902.png" />''
| + | {{TEX|done}} |
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| − | The image (under a translation) of a vector subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048490/h0484903.png" /> with one-dimensional quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048490/h0484904.png" />, i.e. a set of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048490/h0484905.png" /> for a certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048490/h0484906.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048490/h0484907.png" />, the hyperplane is sometimes called homogeneous. A subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048490/h0484908.png" /> is a hyperplane if and only if
| + | ''in a [[vector space]] $X$ over a [[field]] $K$'' |
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048490/h0484909.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
| + | The image (under a translation) of a vector subspace $M$ with one-dimensional quotient space $X/M$, i.e. a set of the form $x_0+M$ for a certain $x_0\in X$. If $x_0=0$, the hyperplane is sometimes called homogeneous. A subset $\pi\subset X$ is a hyperplane if and only if |
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| − | for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048490/h04849010.png" /> and a certain non-zero linear functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048490/h04849011.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048490/h04849012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048490/h04849013.png" /> are defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048490/h04849014.png" /> up to a common factor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048490/h04849015.png" />.
| + | \begin{equation}\label{eq:1} |
| | + | \pi = \{x\colon f(x) = \alpha\} |
| | + | \end{equation} |
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| − | In a topological vector space any hyperplane is either closed or is everywhere dense; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048490/h04849016.png" /> as defined by formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048490/h04849017.png" /> is closed if and only if the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048490/h04849018.png" /> is continuous. | + | for $\alpha\in K$ and a certain non-zero [[linear functional]] $f\in X^*$. Here, $f$ and $\alpha$ are defined by $M$ up to a common factor $\beta\neq 0$. |
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| | + | In a [[topological vector space]] any hyperplane is either [[closed set|closed]] or is everywhere [[dense set|dense]]; $\pi$ as defined by formula \eqref{eq:1} is closed if and only if the [[functional]] $f$ is [[continuous functional|continuous]]. |
Latest revision as of 13:01, 26 April 2016
in a vector space $X$ over a field $K$
The image (under a translation) of a vector subspace $M$ with one-dimensional quotient space $X/M$, i.e. a set of the form $x_0+M$ for a certain $x_0\in X$. If $x_0=0$, the hyperplane is sometimes called homogeneous. A subset $\pi\subset X$ is a hyperplane if and only if
\begin{equation}\label{eq:1}
\pi = \{x\colon f(x) = \alpha\}
\end{equation}
for $\alpha\in K$ and a certain non-zero linear functional $f\in X^*$. Here, $f$ and $\alpha$ are defined by $M$ up to a common factor $\beta\neq 0$.
In a topological vector space any hyperplane is either closed or is everywhere dense; $\pi$ as defined by formula \eqref{eq:1} is closed if and only if the functional $f$ is continuous.
How to Cite This Entry:
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