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Let $E$ be a linear space over a field $k$ . A hyperplane $H$ in $E$ is defined as the set of the form $$H=\{x\in E:f(x)=a\}$$ where $a \in k$ and $f$ is a nonzero linear functional, $f \colon E \to k$ . If $k=\mathbb{R}$ or $\mathbb{C}$ , then $H$ is called a real hyperplane or complex hyperplane respectively.
Remark. When $k=\mathbb{C}$ , the word ``hyperplane'' also has a more restrictive meaning: it is the zero set of a complex linear functional (by setting $a=0$ above).
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"hyperplane" is owned by georgiosl. [ full author list (2) ]
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| Also defines: |
real hyperplane, complex hyperplane |
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Cross-references: complex, zero set, linear functional, field, linear space
There are 12 references to this entry.
This is version 6 of hyperplane, born on 2005-05-10, modified 2008-09-02.
Object id is 7035, canonical name is Hyberplane.
Accessed 3170 times total.
Classification:
| AMS MSC: | 46H05 (Functional analysis :: Topological algebras, normed rings and algebras, Banach algebras :: General theory of topological algebras) |
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Pending Errata and Addenda
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