Definition:Hyperplane

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Definition

Let $X$ be a vector space.

Let $U$ be a proper subspace of $X$.

Definition 1

$U$ is a hyperplane if and only if:

for all subspaces $Z$ of $X$ containing $U$, we have either $Z = U$ or $Z = X$.


Definition 2

$U$ is a hyperplane (in $X$) if and only if $U$ has codimension $1$ in $X$.


Definition 3

$U$ is a hyperplane (in $X$) if and only if:

there exists a non-zero linear functional $\phi : X \to \Bbb F$ such that:
$U = \map \ker \phi$


Also see

  • Results about hyperplanes can be found here.


Sources