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 $x \in X$.

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:

$\ds \map f {i x}$

$=$

$\ds i \map f x$

$\ds $

$=$

$\ds i \paren {\map g x + i \map h x}$

$\ds $

$=$

$\ds i \map g x - \map h x$

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 $x \in X$.

$\blacksquare$