(i) Let $K$ be the operator on $L_2(a, b)$ given by
$$
(K \phi)(x)=\int_a^x \frac{(x-b)(t-a)}{b-a} \phi(t) \mathrm{d} t+\int_x^b \frac{(x-a)(t-b)}{b-a} \phi(t) \mathrm{d} t \quad(a \leqslant x \leqslant b) .
$$
Show that $(K \phi)(a)=(K \phi)(b)=0$ and that $(K \phi)^{\prime \prime}(x)=\phi(x)$ (almost everywhere). By finding the eigenvalues of $K$, show that $\|K\| \leqslant(b-a)^2 /$ $\pi^2$.
(ii) Let $H=L_2(0,1)$ and $E_{n+1}$ be the subspace of functions which are continuous on $[0,1]$ and which are linear on each of the subintervals $[(j-1) / n, j / n](j=1,2, \ldots, n)$. Show that if $\psi$ is twice differentiable and $\psi^{\prime \prime} \in L_2(0,1)$ then $\left\|\psi-P_{n+1} \psi\right\| \leqslant\left\|\psi^{\prime \prime}\right\| /\left(n^2 \pi^2\right)$, where $P_{n+1}$ is the projection of $H$ onto $E_{n+1}$.