= Solution
A <sublinear functional> on a real vector space satisfies $p(x+y)\leq p(x)+p(y)$ and $p(tx)=tp(x)$ for every $t\geq0$. The dominated real <Hahn-Banach theorem> says that a <linear functional> on a subspace satisfying $f\leq p$ extends to a <linear functional> on the whole space still dominated by $p$.
The limit-superior functional is finite on bounded sequences, positively homogeneous, and subadditive, since for every tail $\sup(\alpha_n+\beta_n)\leq\sup\alpha_n+\sup\beta_n$. Passing to the tail limits proves sublinearity. On the subspace $c$ of convergent real sequences, the ordinary limit is linear and equals $p$. Hahn-Banach therefore extends it to $\varphi$ on $\ell^\infty$ with
$$
\boxed{\varphi(x)\leq\limsup_n\alpha_n.}
$$
Back to article page
