= Multiplication operator
{title2=$M_fu=fu$}
{wiki}
For a finite almost-everywhere <measurable function> $f$ on a <sigma-finite measure> space, the multiplication operator on <L2 space> has maximal <operator domain>
$$
D(M_f)=\{u\in L^2:fu\in L^2\},\qquad M_fu=fu.
$$
It is a <closed operator>: if $u_n\to u$ and $fu_n\to v$ in <L2 space>, suitable <subsequences> converge almost everywhere, giving $v=fu$. Its <operator domain> is a <dense subset>, since truncating $u$ to $\{|f|\leq n\}$ approximates it in <L2 space>.
If $f$ is essentially bounded, $M_f$ is a <bounded linear operator> and its <operator norm> is the <essential supremum> of $|f|$. The <upper bound> follows by integrating $|fu|^2\leq\|f\|_\infty^2|u|^2$. For the matching <lower bound>, use a normalized indicator of a finite-positive-measure subset where $|f|$ is within $\epsilon$ of its <essential supremum>; <sigma-finiteness> supplies such a subset. Thus the measure-space hypothesis also justifies the claimed <operator norm>, rather than merely closedness.
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