Real Linear Functional is Real Part of Unique Linear Functional

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

Let $g : X \to \R$ be a $\R$-linear functional.

Then there exists a unique $\C$-linear functional $f : X \to \C$ such that:


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

for each $x \in X$.

Further:


 * $\map f x = \map g x - i \map g {i x}$

Proof
Define $f : X \to \C$ by:
 * $\map f x = \map g x - i \map g {i x}$

for each $x \in X$.

Then for $\lambda, \mu \in \R$ and $x, y \in X$ we have:

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

To show that $f$ is $\C$-linear, we need to show that:
 * $\map f {i x} = i \map f x$

for each $x \in X$.

We have:

Then for $\alpha + i\beta, \gamma + i\delta \in \C$ and $x, y \in X$ we have:

So $f : X \to \C$ is a linear functional with:
 * $\map g x = \map \Re {\map f x}$

Further, from Linear Functional on Complex Vector Space is Uniquely Determined by Real Part, we have that $f : X \to \C$ is unique.