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.
$\blacksquare$