Hahn-Banach Theorem/Complex Vector Space

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

Let $p : X \to \R$ be a seminorm on $X$.

Let $X_0$ be a linear subspace of $X$.

Let $f_0 : X_0 \to \C$ be a linear functional such that:


 * $\cmod {\map {f_0} x} \le \map p x$ for each $x \in X_0$.

Then there exists a linear functional $f$ defined on the whole space $X$ which extends $f_0$ and satisfies:


 * $\cmod {\map f x} \le \map p x$ for each $x \in X$.

That is, there exists a linear functional $f : X \to \C$ such that:


 * $\cmod {\map f x} \le \map p x$ for each $x \in X$

and:


 * $\map f x = \map {f_0} x$ for each $x \in X_0$.