Definition 9.3.1 (Quotient Seminorm). Let $E$ be a vector space over $K \in \RC$, $\rho: E \to [0, \infty)$ be a seminorm, and $M \subset E$ be a vector subspace, then
\[\rho_{M}: E/M \to [0, \infty) \quad x + M \mapsto \inf_{y \in x + M}\rho(y)\]
is the quotient of $\rho$ by $M$.
Proof. Let $\lambda \in K$ with $\lambda \ne 0$, then $x \mapsto \lambda x$ is a bijection, and
\begin{align*}\abs{\lambda}\rho_{M}(x)&= \abs{\lambda}\inf\bracs{\rho(y)|y \in x + M}= \inf\bracs{\abs{\lambda}\rho(y)|y \in x + M}\\&= \inf\bracs{\rho(\lambda y)|y \in x + M}=\inf\bracs{\rho(y)|y \in \lambda(x + M)}= \rho_{M}(\lambda(x + M))\end{align*}
For any $x, x' \in E$, $y \in x + M$ and $y' \in x' + M$, $y + y' \in x + x' + M$, so
\[\rho_{M}(x + x') \le \rho(y + y') \le \rho(y) + \rho(y')\]
As this holds for all $y \in x + M$ and $y' \in x' + M$, $\rho_{M}(x + x') \le \rho_{M}(x) + \rho_{M}(x')$.$\square$