Hyperplane

in a vector space $X$ over a field $K$

The image (under a translation) of a vector subspace $M$ with one-dimensional quotient space $X/M$, i.e. a set of the form $x_0+M$ for a certain $x_0\in X$. If $x_0=0$, the hyperplane is sometimes called homogeneous. A subset $\pi\subset X$ is a hyperplane if and only if

\begin{equation}\label{eq:1} \pi = \{x\colon f(x) = \alpha\} \end{equation}

for $\alpha\in K$ and a certain non-zero linear functional $f\in X^*$. Here, $f$ and $\alpha$ are defined by $M$ up to a common factor $\beta\neq 0$.

In a topological vector space any hyperplane is either closed or is everywhere dense; $\pi$ as defined by formula \eqref{eq:1} is closed if and only if the functional $f$ is continuous.