Consider an operator $T:L^2[0,1]\to L^2[0,1]$ and $Tf(t)=\int_0^t e^{-(t-x)}f(x)dx$. Prove $T$ is bounded and calculate its norm.
To show it is bounded, we need to calculate $\|Tf\|\leq c\|f\|$ for some constant $c$. I have some problems of finding $c$, and calculating its norm, that is, $\|T\|=\sup_{\|f\|=1}\|Tf\|$.