Linear Functional on Complex Vector Space is Uniquely Determined by Real Part

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

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

Define a function $g : X \to \R$:


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

for each $x \in X$.

Then:


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

for each $x \in X$.

Proof
For brevity, define a function $h : X \to \R$ by:


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

for each $x \in X$.

Note that:


 * $\map f x = \map \Re {\map f x} + i \map \Im {\map f x} = \map g x + i \map h x$

so that:


 * $\map f {i x} = \map g {i x} + i \map h {i x}$

for each $x \in X$.

On the other hand, by the linearity of $f$, we have:

for each $x \in X$.

Comparing real parts, we have:


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

so $\map h x = -\map g {i x}$.

So we have:


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

for each $x \in X$.