Hahn–Banach Theorem in Normed Spaces

A bounded linear functional on a subspace extends to the whole space without increasing its norm.

Hahn–Banach Theorem in Normed Spaces

Let $X$ be a normed space , let $Y\subset X$ be a subspace , and let $f:Y\to\mathbb{K}$ be a bounded linear functional .

Theorem (Hahn–Banach, normed spaces): There exists a bounded linear functional $F:X\to\mathbb{K}$ such that

$F|_Y=f$ , and
$\|F\|=\|f\|$ .

Context: This is obtained by applying the seminorm version of Hahn–Banach with the seminorm $p(x)=\|f\|\|x\|$ . It is one of the main tools for constructing supporting functionals and proving geometric separation theorems in normed spaces.