Let V be a real vector space, let p: V o \mathbb{R} be sublinear (Definition ef{def:sublinear_functional_2025_08_19}), let U \subseteq V be a linear subspace, and let f_0: U o \mathbb{R} be linear with f_0(x) \le p(x) for all x \in U. Then there exists a linear functional f: V o \mathbb{R} extending f_0 (i.e., f|U = f_0) such that f(x) \le p(x) for all x \in V.