Real Part of Linear Functional is Linear Functional

Theorem
Let $X$ be a vector space over $\C$.

Let $f : X \to \C$ be a linear functional.

Define $g : X \to \R$ by:


 * $\map g x = \map \Re {\map f x}$

for each $x \in X$.

Then $f$ is $\R$-linear.

Proof
Let $x, y \in X$ and $\lambda, \mu \in \R$.

Then: