Hahn-Banach Theorem/Complex Vector Space/Corollary

Corollary
Let $\struct {X, \norm \cdot}$ be a normed vector space over $\C$.

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

Let $f_0 : X_0 \to \C$ be a bounded linear functional.

Then $f_0$ can be extended to a bounded linear functional $f : X \to \C$ with:


 * $\norm f_{X^\ast} = \norm {f_0}_{\paren {X_0}^\ast}$

where $\norm \cdot_{X^\ast}$ and $\norm \cdot_{\paren {X_0}^\ast}$ are the norms of the normed dual spaces $X^\ast$ and $\paren {X_0}^\ast$.