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 $\tilde f$ defined on the whole space $X$ which extends $f$.

That is:


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

and:


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