I have three linear mappings:
\begin{equation}t_0(f)=f(t_0)\end{equation}
\begin{equation}I(f)=\int_{0}^{1}f(t)f_0(t)dt\end{equation}
\begin{equation}T(f)=f(t)f_0(t)\end{equation}
and I want to determine whether or not they are continuous on $(C[0,1],\|\centerdot\|_1)$.
I have been trying to prove continuity by showing boundedness, e.g. $|T(f)|\leq M\|f\|_1$, with no success. I also have tried to construct a sequence $f_n$ satisfying $\|f_n\|_1=1$ and $|f_n(t)|\rightarrow\infty$, or find a sequence $f_n$ such that $\int_{0}^{1}f_n(t)dt\rightarrow0$ but $|T(f)|\nrightarrow0$. I have a hard time with counter examples.
I would greatly appreciate any hint or push in the right direction.