Originally Posted by

**surjective** Hello,

I am considering the following map:

$\displaystyle T:L^{1}(0,2) \to T:L^{1}(0,2)$, $\displaystyle (Tf)(x)=\int_{0}^{x}tf(t)$

I wan't to show that $\displaystyle L^{1}(0,2)$ maps to $\displaystyle L^{1}(0,2)$. I have done the following (but get stuck):

$\displaystyle \int_{0}^{2}|(Tf)(x)|dx=\int_{0}^{2}\vert \int_{0}^{x}tf(t)dt \vert dx \leq \int_{0}^{2} \int_{0}^{x}|tf(t)|dt dx$

How should I continue so that I and up with an expression which shows that $\displaystyle \int_{0}^{2}|(Tf)(x)|dx<\infty$? Swich of integral?

Thanks.