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, $f \colon E \to k$ . If $k=\mathbb{R}$ or $\mathbb{C}$ , then $H$ is called a real hyperplane or complex hyperplane respectively.

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