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 $|| \ ||_1 \text{ and } \ || \ ||_{[0.1]}$ equivalent.

Question

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 $|| \ ||_1 \text{ and } \ || \ ||_{[0.1]}$ equivalent.

zzr0ck3r · Answer

$\int_0^1|f(t)| dt\le \int_0^1\max_{t\in[0,1]}|f(t)|dt=\max_{x\in[0,1]}|f(x)|=||f||_{[0,1]}$

Is this all I need to do @eliassaab 

For part b) I think they are not, do I just need to find a function to show they are not?

zzr0ck3r · Answer

i found one:) this is all done:)

anonymous · Answer

Good job