= Peano kernel norm
{c}
{title2=$|L(u)|\leq\|K\|_{L^1}\|u^{(r)}\|_\infty$}
The <Peano kernel theorem> writes a derivative-based error functional as $L(u)=\int K(t)u^{(r)}(t)\,dt$, so its bound in the displayed derivative <seminorm> is at most the <L1 norm> of the kernel. This is the sharp constant: continuous derivatives can approximate the sign of $K$ in the weighted integral, and integrating them $r$ times produces admissible test functions. If $K$ has one sign, a constant $r$th <derivative> already attains equality; in particular $\int K=L[x^r/r!]$. This provides a direct exact check on proposed error constants before estimating individual polynomial pieces.
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