Let X be a real vector space. A map p:X→R is called \emph{sublinear} if (i) p(x+y)≤p(x)+p(y) for all x,y∈X, and (ii) p(λx)=λp(x) for all x∈X and all scalars λ≥0.
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.
Let V be a real vector space, p: V o \mathbb{R} sublinear, U \subseteq V a linear subspace, and f: U o \mathbb{R} linear with f \le p on U. For any v_0 \in V \setminus U, define\alpha := \sup{x \in U} \{\, f(x) - p(x+v_0) \,\},\qquad eta := \inf_{x \in U} \{\, p(x - v_0) - f(x) \,\}.Then \alpha \le eta. For any a \in [\alpha,eta], the formulaildef(x+tv0):=f(x)+ta(x∈U,t∈R)extends f to the subspace U \oplus \mathbb{R} v_0 and satisfies ildef≤p on U \oplus \mathbb{R} v_0.
Let V be a real vector space. A function p: V o \mathbb{R} is called \emph{sublinear} if (i) p(x+y) \le p(x)+p(y) for all x,y \in V, and (ii) p(\lambda x) = \lambda,p(x) for all x \in V and all \lambda \ge 0.
For all vectors u and v in an inner product space, it always holds that \left|\langle u, v \rangle\right|^2 \leq \langle u, u \rangle \cdot \langle v, v \rangle.