Let $X$ be a real vector space, let $M\subseteq X$ be a linear subspace, and let $p: X\to\mathbb{R}$ be sublinear in the sense of \ref{def:sublinear_2025_08_19}. If $f: M\to\mathbb{R}$ is a linear functional (\ref{def:linear_functional_2025_08_19}) such that $f(x)\le p(x)$ for all $x\in M$, then there exists a linear functional $F: X\to\mathbb{R}$ extending $f$ (i.e., $F\vert_M=f$) with $F(x)\le p(x)$ for all $x\in X$.