Hahn–Banach Theorem in Normed Spaces

Positive definiteness: ∥ x ∥ = 0 \|x\|=0 ∥ x ∥ = 0 iff x = 0 x=0 x = 0 .
Absolute homogeneity: ∥ α x ∥ = ∣ α ∣ ∥ x ∥ \|\alpha x\|=|\alpha|\,\|x\| ∥ α x ∥ = ∣ α ∣ ∥ x ∥ .
Triangle inequality: ∥ x + y ∥ ≤ ∥ x ∥ + ∥ y ∥ \|x+y\|\le \|x\|+\|y\| ∥ x + y ∥ ≤ ∥ x ∥ + ∥ y ∥ .

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
$\|F\|=\|f\|$ .

Remarks

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.