I've been trying to prove that in $C[a,b]$ there is no C such that $Cd_{\infty}(f,g))\leq d_{1}(f,g)$ with $d_{\infty}(f,g)=\sup {|f(x)-g(x)|}$ and $d_{1}(f,g)=\int_{a}^{b} |f(x)-g(x)|dx$. And well, proving that and proving $i_{1,\infty}:(C[a,b],d_{1}) \rightarrow (C[a,b],d_{\infty})$ is not continuous is the same so I proved that for the $C[0,1]$ interval which is pretty easiest to prove, but I haven't been able to prove it in $C[a,b]$ for any $a,b\in \mathbb{R} $, could you guys help me out?
EDIT: I will be more specific, I want to prove that for $i_{1,\infty}:(C[a,b],d_{1}) \rightarrow (C[a,b],d_{\infty})$ no matter what $\delta$ I choose, if I fix $\varepsilon=1/2$, $B_{1}(f_{1},\delta) \not\subset B_{\infty}(f_{1},1/2)$ with $f_{1}$ some 'usefull' function in $C[a,b]$
Thanks so much.
For $n\in \Bbb N$ let $f_n$ be linear on $[0,1/2n]$ and on $[1/n,1]$ with $f(0)=f(1/n)=0$ and $f(1/2n)=1.$ Let $f(x)=0$ for $x\in [1/n,1].$
For $n\in N$ we have $d_1(f_n,0)=1/2n$ and $d_{\infty}(f_n,0)=1.$
So there is no $positive$ $ C $ such that $Cd_{\infty}(f_n,0)\leq d_1(f_n,0)$ for all $n\in \Bbb N.$
The $d_{\infty}$ open ball $B_{\infty}(0,1)$ of radius $1,$ centered at the function $0,$ contains no member of $\{f_n:n\in \Bbb N\}.$ So there is no $r>0$ such that the $d_1$ open ball $B_1(0,r)$ is a subset of $B_{\infty}(0,1)$ because $B_1(0,r)\supset \{f_n: 2n>1/r\}\ne \emptyset. $