TITLE: Corollary of Hahn-Banach theorem
QUESTION [3 upvotes]: Let E be a normed linear space. Let $G \subset E$ a linear subspace. Show that if $g : G \to \mathbb{R}$ is a continuous linear operator, then $\exists f \in E^*$ such that $f_{/G} = g$ and $\|f\|_{E^*} = \|g\|_{G^*}$. 
Let $p : E \to \mathbb{R}, p(x) := \|g\|_{G^*}\|x\|, \forall x \in E$ . Then
$$g(x) <p(x), \forall x \in G.$$
From Hahn-Banach Theorem we have that 
$\exists f : E \to \mathbb{R}$ linear such that
$$g(x) = f(x), \forall x \in G$$  and
$$f(x) \leq p(x), \forall x \in E.$$
I can't show that $\|f\|_{E^*} = \|g\|_{G^*}$.
Thank you!

REPLY [2 votes]: Since $f(x) \leq p(x)$, $\| f \| \leq \| g \|$ by the definition of $p$. On the other hand, $\| f \| \geq \| g \|$ is essentially trivial because $\sup \{ f(x) : x \in E, \| x \| = 1 \} \geq \sup \{ f(x) : x \in G,\| x \| = 1 \}$. In other words an extension can only increase the norm or keep it the same.
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