Analysis II | Part IB, 2001

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

d1(x,y)=supt[a,b]x(t)y(t)d2(x,y)={abx(t)y(t)2dt}1/2\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 d1d_{1} metric? Justify your answer.

Show that the set of functions

xn(t)={0,1t<0nt,0t<1/n1,1/nt1x_{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 d2d_{2} metric on C[1,1]C[-1,1], yet does not tend to a limit in the d2d_{2} metric in this space. Hence, deduce that this space is not complete in the d2d_{2} metric.

Typos? Please submit corrections to this page on GitHub.