Hahn–Banach Theorem (Real Vector Spaces)

A linear functional dominated by a sublinear function extends to the whole space.

Hahn–Banach Theorem (Real Vector Spaces)

Hahn–Banach (real version): Let $X$ be a real vector space , let $Y\subset X$ be a linear subspace , and let $p:X\to\mathbb{R}$ be sublinear .

If $f:Y\to\mathbb{R}$ is a linear functional satisfying

f(y)\le p(y)\quad\text{for all }y\in Y,

then there exists a linear functional $F:X\to\mathbb{R}$ such that

$F(y)=f(y)$ for all $y\in Y$ , and
$F(x)\le p(x)$ for all $x\in X$ .

Context: This extension theorem is the analytic backbone of convex separation results (e.g., geometric Hahn–Banach separation ). In the notes, the proof uses Zorn’s lemma to extend $f$ maximally and then shows the domain must be all of $X$ .