# Thread: does this define a norm on R[0,1]?

1. ## does this define a norm on R[0,1]?

R[0,1] is the vector space of all integrable functions on [0,1]

$\|f\|_{1}=\int_{0}^{1}{|f|}$

does this define a norm?

i think yes. because i cant find any axiom fails

2. Hi,

How did you show that $\|f\|_1=0$ implies $f=0$ ?

3. ## ob

isnt it obvious?

u just show the upper riemann sum=lower riemann sum=0 ( by integrale)

therefore, sup f=inf f=0

4. Originally Posted by szpengchao
u just show the upper riemann sum=lower riemann sum=0 ( by integrale)

therefore, sup f=inf f=0
This doesn't work for $
f:x \mapsto \begin{cases} 0 &\mathrm{if}\,\,0\leq x<1 \\ 1& \mathrm{if}\,\, x=1\\ \end{cases}$
since one has $f\neq 0$ but $\int_0^1|f(x)|\,\mathrm{d}x=0$.