\(C(\infty,\infty)\):  The Axiom of Choice: Every  set  of  non-empty sets has a choice function.

Generalized Hahn-Banach Theorem:  Assume that \(X\) is a real vector space, \((Z,\preccurlyeq)\) is a Dedekind complete ordered vector space and \(X_0\) is a subspace of \(X\).  If \(\lambda_0 : X_0 \to Z\) is linear and \(p: X\to Z\) is sublinear and if \(\lambda_0 \preccurlyeq p\) on \(X_0\) then \(\lambda_0\) can be extended to a linear map \(\lambda : X\to Z\) such that \(\lambda \preccurlyeq p\) on \(X\). \ac{Schechter} \cite{1996b}