Got stuck in this problem:
Let us denote by $X$ the linear space $C^1([0,1])$ equipped with the norm of $C^0([0,1])$ and consider the following statement:
"The differentiation operator $L:X \rightarrow C^0([0,1]): u \mapsto u'$ is linear and closed. By the Closed Graph Theorem, the operator is then continuous"
Show that the conclusion is wrong and find the mistake in the argument
To show that the conclusion is wrong it should be enough to show that the operator is not bounded (as it is obviously linear and, as it is linear, it is continuous iff it is bounded), which is not hard.
So I believe the mistake is that $L$ is not closed, but how can I prove it? I was trying to find an example of a closed set in $X$ whose image is not closed but I'm not sure whether this is a good approach...