\begin{definition}[Definition:Hyperplane]
Let $X$ be a vector space. 
Let $U$ be a proper subspace of $X$. 
We say that $U$ is a '''hyperplane''' {{iff}}:
:for all subspaces $Z$ of $X$ containing $U$, we have either $Z = U$ or $Z = X$.
That is, {{iff}} $U$ has codimension $1$.
\end{definition}