Problem :
Let $E=\left(\mathcal{C}[0,1],\|.\|_{\infty}\right)$
Defined operator $A$ :
$$A ~~:E\to~~E$$ $$Af(x)=\displaystyle\int\limits_{0}^{x}f(t)dt$$
- Prove that $\|A\|=1$
Then $A^{-1}$ existed , is bounded??
My attempts:
I was find $\|A\|\leq 1$
For $f≡1$ then $\|f\|_{\infty}=1$
But I can't get $\|Af\|_{\infty}=1$
The second question I need to prove $A$ are invective so ,
$Af(x)=0\implies \displaystyle\int\limits_{0}^{x}f(t)dt=0$
$\implies f(t)=0 $
by derivative both side is this correct??
And what about bounded of $A^{-1}$