Let $|| \ ||_1$ be the norm on $C([0,1])$ defined by $||f||_1 = \int_0^1|f(t)|dt$.
a) Show that $||f||_1 \le ||f||_{[0,1]}$
b) Are $|| \ ||_1$ and $|| \ ||_{[0,1]}$ equivalent?
For part a) I think this works, but any help would be great if there are glaring problems
$\int_0^1|f(t)| dt\le \int_0^1\max_{t\in[0,1]}|f(t)|dt=\max_{x\in[0,1]}|f(x)|=||f||_{[0,1]}$
And for part b) I am thinking its wrong (or they would have said prove it) so do I just need to find a function where the two norms give different values?
Hint: consider a piecewise function $f_n$, taking the value $n$ in $0$, affine on $(0,1/n^2)$, and taking the value $0$ from $1/n^2$ to $1$.
$\|f_n\|_1 = 1/2n, \|f\|_{[0,1]} = n$.