Imaginary 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 $h : X \to \R$ by:

$\map h x = \map \Im {\map f x}$

for each $x \in X$.

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

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

Then:

$\ds \map h {\lambda x + \mu y}$

$=$

$\ds \map \Re {\map f {\lambda x + \mu y} }$

$\ds $

$=$

$\ds \frac 1 {2 i} \paren {\map f {\lambda x + \mu y} - \overline {\map f {\lambda x + \mu y} } }$

$\ds $

$=$

$\ds \frac 1 {2 i} \paren {\lambda \map f x + \mu \map f y - \overline {\lambda \map f x} - \overline {\mu \map f y} }$

$\ds $

$=$

$\ds \frac 1 {2 i} \paren {\lambda \map f x + \mu \map f y - \lambda \overline {\map f x} - \mu \overline {\map f y} }$

$\ds $

$=$

$\ds \frac \lambda {2 i} \paren {\map f x - \overline {\map f x} } + \frac \mu {2 i} \paren {\map f y - \overline {\map f y} }$

$\ds $

$=$

$\ds \lambda \map g x + \mu \map g y$

$\blacksquare$