Definition 4.1.4 (Seminorm).label Let $E$ be a vector space over $K\in \curl{\R,\com}$, then a seminorm on $E$ is a mapping $\rho:E\to[0,\infty)$ such that:

1. (SN1)

   $\rho(0)=0$

2. (SN2)

   $\fall x\in E,\fall \lam\in K,\rho(\lam x)=\abs{\lam}\rho(x)$

3. (SN3)

   $\fall x,y\in E,\rho(x+y)\leq \rho(x)+\rho(y)$