metrics of function space - supermum and definite integral induce different topology?

Question

Given $C[0,1]$ (continuous function space in $[0, 1]$), Do supermum and definite integral as metrics induce a different topology?

Answer 1

If you are talking about the metrics $d(f,g)=\sup \{|f(x)-g(x)|: 0 \leq x \leq 1\}$ and $d'(f,g)=\int_0^{1} |f(x)-g(x)|\, dx$ then they are not equivalent: if $f_n(x)=1-nx$ for $ 0\leq x \leq \frac 1 n$ and $f_n(x)=0$ for $x >\frac 1 n$ then $f_n \to 0$ in the metric $d'$ but not in $d$.