Definition 9.1.6 (Seminorm). Let $E$ be a vector space over $K \in \RC$, then a seminorm on $E$ is a mapping $\rho: E \to [0, \infty)$ such that:

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

2. For any $x \in E$ and $\lambda \in K$, $\rho(\lambda x) = \abs{\lambda}\rho(x)$.

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