Category:Hyperplanes

This category contains results about hyperplanes.
Definitions specific to this category can be found in Definitions/Hyperplanes.

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:

$U = \map \ker \phi$