= Seminorm
{title2=$p$}
{wiki}
A seminorm on a real or complex <vector space> $E$ is a finite-valued <function> $p:E\to[0,\infty)$ satisfying absolute homogeneity and the <triangle inequality>:
$$
p(\lambda x)=|\lambda|p(x),\qquad p(x+y)\leq p(x)+p(y)
\qquad(x,y\in E,\ \lambda\in\mathbb K).
$$
In particular $p(0)=0$ and $p(-x)=p(x)$. A <seminorm> is a <norm> exactly when $p(x)=0$ implies $x=0$: this is the extra <definiteness of a norm> axiom. The zero <function> is a valid <seminorm>. The missing definiteness is useful when a measurement deliberately ignores some directions, such as an individual coordinate or the constant term of a <function> whose <derivative> is measured.
For example, on $C^1([0,1])$ the <function> $p(f)=\sup_{t\in[0,1]}|f'(t)|$ is a <seminorm> that vanishes on every constant <function>. Adding the value measurement gives the <norm> $q(f)=|f(0)|+p(f)$: if $q(f)=0$, the <fundamental theorem of calculus> gives $f(t)=f(0)+\int_0^t f'(s)\,ds=0$. A family of such partial measurements defines a <seminorm topology>.
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