Let $T$ be a linear map from $L_p[0,1]$ to $L_p[0,1]$, $1\leq p < \infty$. If $(f_n)$ converges to $0$ almost everywhere, then $(T(f_n))$ converges to $0$ almost everywhere. How does this imply that $T$ is bounded?
I have been trying to take a sequence $f_n$ converging to $0$ in $L_p$, then taking a subsequence $f_{n_k}$which is a.e convergent to $0$. Now, $Tf_{n_k}$ converges to $0$ a.e. Now,if I can obtain a dominating function for $Tf_{n_k}$, I could use the dominated convergence theorem.
However, this still would not complete the proof, as this only shows there is a subsequence of $Tf_n$ converging to $0$ in $L_p$.
Any hints?
As PhoemueX said, the closed graph theorem helps. According to it, you only need to know that if $f_n\to 0$ in $L^p$ and $Tf_n\to g$ in $L^p$, then $g=0$.
This follows from the condition given: indeed, there is a subsequence $f_{n_k}$ converging to $0$ a.e., so $Tf_{n_k}\to 0$ a.e., so $g=0$.