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.

Answer 1

\(\int_0^1|f(t)| dt\le \int_0^1\max_{t\in[0,1]}|f(t)|dt=\max_{x\in[0,1]}|f(x)|=||f||_{[0,1]}\)

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?

Answer 2

i found one:) this is all done:)

Answer 3

Good job