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Let $X$ be a vector space.
Let $U$ be a proper subspace of $X$.
$U$ is a hyperplane if and only if:
- for all subspaces $Z$ of $X$ containing $U$, we have either $Z = U$ or $Z = X$.
$U$ is a hyperplane (in $X$) if and only if $U$ has codimension $1$ in $X$.
$U$ is a hyperplane (in $X$) if and only if:
- there exists a non-zero linear functional $\phi : X \to \Bbb F$ such that:
- $U = \map \ker \phi$
- Results about hyperplanes can be found here.
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): hyperplane