Definition 11.1.8 (Seminorm).label 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. (SN1)

   $\rho(0) = 0$.

2. (SN2)

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

3. (SN3)

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