Hahn-Banach Theorem/Real Vector Space/Corollary 1

Corollary
Let $X$ be a vector space over $\R$. Let $p : X \to \R$ be a seminorm on $X$.

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

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


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

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

That is:


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

and:


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

Proof
From Seminorm is Sublinear Functional, we have:


 * $p$ is a sublinear functional.

So, from Hahn-Banach Theorem: Real Vector Space, there exists an extension $f$ with:


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

Then, we have:

So we also have:


 * $-\map f x \le \map p x$ for each $x \in X$.

So:


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

and so $f$ is the desired extension.