= Kernel of a seminorm
{title2=$\ker p=\{x:p(x)=0\}$}
= Kernels of seminorms
{synonym}
The kernel of a <seminorm> is its zero set, a <linear subspace> by absolute homogeneity and the <triangle inequality>. The estimate $|p(x)-p(y)|\leq p(x-y)$ shows that adding an element of $\ker p$ does not change $p$. Hence $\|x+\ker p\|=p(x)$ defines a <norm> on the <quotient vector space> $E/\ker p$.
On $E$ itself, $d_p(x,y)=p(x-y)$ is a <pseudometric>; it is a <metric> exactly when $p$ is a <norm>. In particular, zero distance in one <pseudometric> need not mean equality of <vectors>. A <separating family of seminorms> removes the common zero-distance ambiguity.
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