Show that each of the functions below is a metric on the set of functions $x(t) \in$ $C[a, b]$ :

$$\begin{gathered} d_{1}(x, y)=\sup _{t \in[a, b]}|x(t)-y(t)| \\ d_{2}(x, y)=\left\{\int_{a}^{b}|x(t)-y(t)|^{2} d t\right\}^{1 / 2} \end{gathered}$$

Is the space complete in the $d_{1}$ metric? Justify your answer.

Show that the set of functions

$$x_{n}(t)= \begin{cases}0, & -1 \leqslant t<0 \\ n t, & 0 \leqslant t<1 / n \\ 1, & 1 / n \leqslant t \leqslant 1\end{cases}$$

is a Cauchy sequence with respect to the $d_{2}$ metric on $C[-1,1]$, yet does not tend to a limit in the $d_{2}$ metric in this space. Hence, deduce that this space is not complete in the $d_{2}$ metric.