hyperplane

Let $E$ be a linear space over a field $k$. A hyperplane $H$ in $E$ is defined as the set of the form

$$H=\{x\in E:f(x)=a\}$$

where $a\in k$ and $f$ is a nonzero linear functional, $f:E\to k$. If $k=ℝ$ or $ℂ$, then $H$ is called a real hyperplane or complex hyperplane respectively.

Remark. When $k=ℂ$, the word “hyperplane” also has a more restrictive meaning: it is the zero set of a complex linear functional (by setting $a=0$ above).