Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 319 / 1 / b / Solution

For $x\in X$ and $t>0$, form the Bochner integralThe semigroup property gives, for $h>0$,Strong continuity lets $h\downarrow 0$, yieldingAlso,by strong continuity. Every $x\in X$ is therefore a norm limit of elements of $D(A)$, soThis approximation is the basic Yosida averaging of a semigroup.