= Solution
Let $(u_n)$ be a <Cauchy sequence> in the <graph norm>
$$
\|u\|_Y=\|u\|+\|Au\|.
$$
Then $(u_n)$ and $(Au_n)$ are <Cauchy sequences> in the <Banach space> $H$, so for some $u,v\in H$,
$$
u_n\to u,
\qquad
Au_n\to v.
$$
Because $A$ is a <closed linear operator>, $u\in D(A)$ and $Au=v$. It follows that
$$
\|u_n-u\|_Y=\|u_n-u\|+\|Au_n-Au\|\longrightarrow0.
$$
Thus every <Cauchy sequence> converges in $Y=(D(A),\|\cdot\|_Y)$, and
$$
\boxed{Y\text{ is a Banach space}}.
$$
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