 Solution

Let $(u_n)$ be a Cauchy sequence in the graph normThen $(u_n)$ and $(A{u}_n)$ are Cauchy sequences in the Banach space $H$, so for some $u,v\in H$,Because $A$ is a closed linear operator, $u\in D(A)$ and $Au=v$. It follows thatThus every Cauchy sequence converges in $Y=(D(A),\Vert \cdot {\Vert }_Y)$, and