Proposition 9

Let $E$ be a normed space over a non-discrete valued division ring $K$, and let $H$ be a closed vector subspace of $E$. If, for each class $\dot{x} \in E/H$, we put $||\dot{x}|| = \inf_{x \in \dot{x}} ||x||$, the function $||\dot{x}||$ is a norm on the vector space $E/H$, and the topology defined by this norm is the quotient by $H$ of the topology of $E$.

By Remark 2 of no. 1, $d(\dot{x}, \dot{y}) = ||\dot{x} - \dot{y}||$ is an invariant metric on $E/H$ which defines the quotient by $H$ of the topology of $E$. It remains only to show that $||t \dot{x}|| = |t| \cdot ||\dot{x}||$, and this follows immediately from the definition of $||\dot{x}||$ [Chapter IV, § 5, no. 7, formula (23)].

The norm $||\dot{x}||$ may also be interpreted as follows: it is the distance (in $E$) of every point $x \in \dot{x}$ from the subspace $H$, for the points of $\dot{x}$ are the points $x - z$, where $z$ runs through $H$.