= Sandwich form of the Hahn-Banach theorem
{title2=$q\leq U\leq p$}
If $p$ is a <sublinear functional>, $q$ a <superlinear functional> and $q\leq p$ on a real vector space, there exists a linear functional $U$ between them. More generally, a linear functional $S$ on a subspace extends with this domination whenever $S(y)\leq p(x+y)-q(x)$ for all ambient $x$ and subspace $y$. In a one-dimensional extension the <admissible values for a sandwich extension> form a nonempty real interval. Positive homogeneity checks the extension on both signs of the new coordinate. Chain unions preserve linearity and the mixed domination, so <Zorn's lemma> gives a maximal extension, whose domain must be the whole space. Finally $x=0$ in the mixed inequality gives the upper bound; using the pair $(x,-x)$ gives the lower bound. Conversely, any linear $U$ between $q,p$ obeys the mixed inequality by $U(y)=U(x+y)-U(x)$.
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