(i) Let $K$ be the operator on $L_2(a, b)$ defined by
$$
(K \phi)(x)=\int_a^x \frac{t-a}{b-a} \phi(t) \mathrm{d} t+\int_x^b \frac{t-b}{b-a} \phi(t) \mathrm{d} t \quad(a \leqslant x \leqslant b) .
$$
Show that, for $\phi \in L_2(a, b), \int_a^b(K \phi)(x) \mathrm{d} x=0$ and $(K \phi)^{\prime}(x)=\phi(x)$ (almost everywhere). Show also that $\|K\| \leqslant(b-a) / 6^{\frac{1}{2}}$.
Let $E$ be the subspace of $L_2(a, b)$ consisting of constant functions and $P$ be the projection of $L_2(a, b)$ onto $E$. Show that if $\psi$ is differentiable and $\psi^{\prime} \in L_2(a, b)$, then $\psi-P \psi=K\left(\psi^{\prime}\right)$, and thus $\|\psi-P \psi\| \leqslant(b-a)\left\|\psi^{\prime}\right\| / 6^{ \pm}$.
(ii) Let $H=L_2(0,1)$ and $E_n$ be the subspace consisting of those functions constant on each of the intervals $((j-1) / n, j / n)(j=1,2, \ldots, n)$. Show that if $P_n$ is the projection of $H$ onto $E_n$ then, for every differentiable function $\psi$ whose derivative belongs to $L_2(0,1),\left\|\psi-P_n \psi\right\| \leqslant\left\|\psi^{\prime}\right\| /\left(n(6)^{\frac{1}{2}}\right)$.