= Solution
Take the <normed space> $X=C^1([-1,1])$ with the <supremum norm>, and let $T:X\to C([-1,1])$ be the inclusion $Tf=f$. This is an injective <linear map>, so its <kernel> is the <complete> zero space. Its <graph norm> is
$$
\boxed{\|f\|_T=2\|f\|_\infty.}
$$
The <Cauchy sequence> $f_n(x)=\sqrt{x^2+n^{-2}}$ from part (d)(ii) still has no limit in $X$ for this <norm>, because its only possible limit is $|x|$. Consequently
$$
\boxed{\ker T=\{0\}\text{ is complete, but }(X,\|\cdot\|_T)\text{ is incomplete}.}
$$
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