📂Functions

Semi-Linear (Conjugate Linear) Functions

Definition

Assuming that a function $f : X \to \mathbb{C}$ is given. If the following equation holds for $x,y\in X$, $a,b \in \mathbb{C}$, then $f$ is called antilinear or conjugate linear.

$$f(ax + by)=\overline{a}f(x)+\overline{b}f(y)$$

Explanation

Unlike linear functions, where the multiplied constant is the same inside and outside the function, it refers to a function in which the constant is the conjugate complex number inside and outside the function. If $a \in \mathbb{R}$, then because $a=\overline{a}$, a function being real-valued and linear is the same as it being antilinear.