Let $(V,\lVert\mathord{\cdot}\rVert)$ be a seminormed vector space over $k$. If $W$ is a vector subspace of $V$, then map $(x\in W)\mapsto\lVert x\rVert$ defines a seminorm on $W$, called the restriction of $\lVert\mathord{\cdot}\rVert$ on $W$. If $Q$ is a quotient vector space of $V$ and $\pi:V\rightarrow Q$ is the quotient map, then the map $(q\in Q)\mapsto\inf_{x\in\pi^{-1}(\{q\})}\lVert x\rVert$ defines a seminorm on $Q$, called the quotient of $\lVert\mathord{\cdot}\rVert$ on $Q$.