Continuous Real Linear Functional on Complex Topological Vector Space is Real Part of Continuous Complex Linear Functional

Theorem

Let $\struct {X, \tau}$ be a topological vector space over $\C$.

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

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

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

for each $x \in X$.

Proof

From Real Linear Functional is Real Part of Unique Linear Functional, there exists a unique $\C$-linear functional $f : X \to \C$ such that:

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

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

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

We need to show that $g$ is continuous.

From the definition of a topological vector space, the map $c : \C \times X \to X$ defined by:

$\map c {\lambda, x} = \lambda x$

for each $\tuple {\lambda, x} \in \C \times X$ is continuous.

From Vertical Section of Continuous Function is Continuous, the map $c_i : X \to X$ defined by:

$\map {c_i} x = i x$

for each $x \in X$ is continuous.

From Composite of Continuous Mappings is Continuous:

$x \mapsto \map g {i x}$ is continuous.

From Linear Combination of Continuous Functions valued in Topological Vector Space is Continuous, we have that $f$ is continuous.

$\blacksquare$