Bounded Real-Valued Linear Functional is Real Part of Unique Bounded Complex-Valued Linear Functional

Theorem
Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space over $\C$.

Let $\struct {X_\R, \norm {\, \cdot \,}_\R}$ be the realification of $X$ equipped with the restricted norm.

Let $f : X_\R \to \R$ be a bounded linear functional.

Then there exists a unique bounded linear functional $g : X \to \C$ such that:


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

for each $x \in X$.

Proof
Let $M > 0$ be such that:


 * $\cmod {\map f x} \le M \norm x$

for each $x \in X$.

From Linear Functional on Complex Vector Space is Uniquely Determined by Real Part, there exists a unique linear functional $g : X \to \C$ such that:


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

for each $x \in X$, given by:


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

for each $x \in X$.

We just need to show that $g$ is bounded.

Then we have, for each $x \in X$:

so that:


 * $\cmod {\map g x} \le M \sqrt 2 \norm x$

for each $x \in X$.

So $g$ is a bounded linear functional as required.