= Generator of a strongly continuous semigroup is closed and densely defined
The <Yosida averaging of a semigroup> gives $x_h=h^{-1}\int_0^hS(t)x\,dt\in D(A)$ with $x_h\to x$, proving that the <generator domain> is dense. For $x\in D(A)$, the <Bochner integral> identity $S(t)x-x=\int_0^tS(s)Ax\,ds$ holds. If $x_n\to x$ and $Ax_n\to y$, pass to the limit in this identity and divide by $t$. The <strongly continuous semigroup> property makes the resulting limit $y$, so $x\in D(A)$ and $Ax=y$. This proves that the <infinitesimal generator of a semigroup> is a <closed operator>.
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