One-dimensional dominated extension of a real linear functional (${\sup }_m(g(m)-p(m-v))\le c\le {\inf }_m(p(m+v)-g(m))$)

A real linear functional $g$ on a vector subspace, dominated by a sublinear function $p$, can be extended across one new vector $v$ by setting $\tilde{g}(m+tv)=g(m)+tc$. The displayed lower and upper bounds are compatible because $g(m+n)\le p(m-v)+p(n+v)$. A value between them preserves domination for both signs of $t$. Chain unions and Zorn's lemma then yield the full Hahn-Banach theorem.