= Solution
The <kernel> $W=\langle1\rangle$ is <complete>: the map $a\mapsto a1$ is an <isometry> from $\mathbb R$ with its absolute-value <norm> onto $W$ with the <supremum norm>.
Define a <linear map> $S:Y\to X$ by
$$
(Sg)(x)=\int_0^x g(t)\,dt.
$$
The <fundamental theorem of calculus> gives $Sg\in C^1([-1,1])$ and $TSg=g$. Also,
$$
|(Sg)(x)|\leq|x|\|g\|_\infty\leq\|g\|_\infty,
\qquad\|Sg\|_X\leq\|g\|_Y.
$$
Thus $S$ is a bounded <right inverse>, and $Y=C([-1,1])$ is a <Banach space>. Applying part (c),
$$
\boxed{\bigl(C^1([-1,1]),\|f\|_\infty+\|f'\|_\infty\bigr)\text{ is complete}.}
$$
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