Definition 9.1.5 (Sublinear Functional). Let $E$ be a vector space over $K \in \RC$, then a sublinear functional is a mapping $\rho: E \to \real$ such that:

1. $\rho(0) = 0$.

2. For any $x \in E$ and $\lambda \ge 0$, $\rho(\lambda x) = \lambda \rho(x)$.

3. For any $x, y \in E$, $\rho(x + y) \le \rho(x) + \rho(y)$.