Hahn-Banach Theorem/Real Vector Space/Corollary 1

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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:

$\forall x \in X: \map f x \le \map p x$

Then, we have:

\(\ds -\map f x\) \(=\) \(\ds \map f {-x}\) Definition of Linear Functional
\(\ds \) \(\le\) \(\ds \map p {-x}\)
\(\ds \) \(=\) \(\ds \size {-1} \map p x\) Definition of Seminorm
\(\ds \) \(=\) \(\ds \map p x\)


So we also have:

$\forall x \in X: -\map f x \le \map p x$

So:

$\forall x \in X: \size {\map f x} \le \map p x$

and so $f$ is the desired extension.

$\blacksquare$