Show that the closed unit ball $B[0,1]$ in $C[0,1]$ is not compact under the following metrics:
$1. d(f,g)=\sup_{x\in [0,1]}|f(x)-g(x)|$
$2.d(f,g)=\int _0^1 |f(x)-g(x)| dx$
My try:
In order to show not compact if we can find a sequence which has no convergent subsequence then we are done.
How to approach $1,2$?
Best Answer
Use functions that are $n^2$ on $[0,1/n]$ and zero on $[1/n+\epsilon,1]$ and on $[1/n,1/n+\epsilon]$ connect them to be continuous. Then the integral of any one of them is close to $n$ (choose an appropriate $\epsilon$ that will depend on $n$) and I believe the integral of the difference of any two of them is at least one.