= Generator of a multiplication semigroup
{title2=$Lf=-kf,\quad D(L)=\{f:kf\text{ belongs to the underlying space}\}$}
The <infinitesimal generator of a semigroup> for $P_tf=e^{-tk}f$ is multiplication by $-k$ on its maximal domain. The bound $|(1-e^{-tk})/t|\leq k$ proves the <norm> <derivative> when $kf$ is integrable; in $C_0$ use a compact-small-tail argument on $kf$. Conversely <norm> convergence identifies the pointwise <derivative> as $-kf$. Limits of graph pairs satisfy the same multiplication identity, proving it is a <closed operator>.
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