Let $V$ be a real vector space, let $p:V\to\mathbb{R}$ be sublinear (Definition \ref{def:sublinear_2025_8_19}), let $U\subseteq V$ be a linear subspace, and let $f_0:U\to\mathbb{R}$ be linear with

Then there exists a linear $f:V\to\mathbb{R}$ such that $f|_U=f_0$ and