Hahn–Banach Extension Dominated by a Seminorm (Real Case)

Let $X$ be a real vector space, let $Y\subset X$ be a subspace, and let $p:X\to\mathbb{R}$ be a seminorm.

Theorem. If $f:Y\to\mathbb{R}$ is linear and satisfies

|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

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

Remarks

This is obtained from the sublinear Hahn–Banach theorem by applying it to both $f$ and $-f$ (or, equivalently, to the sublinear function $x\mapsto p(x)$ with symmetric domination).