So suppose we look at the operator $S:H_0^1(0,1) \rightarrow L^2(0,1), \ u\mapsto u' $,
where $H_0^1(0,1)$ denotes the closure of the infinitely differentiable functions compactly supported in $(0,1)$ in $H^1(0,1)$ and $u'$ the weak derivative of $u$. Is this operator closed? I remember that one can show that the operator $A:C^1[0,1] \rightarrow C[0,1] $ is closed by using a theorem for a sequence of differentiable functions, where under certain conditions the limit of the sequence was differentiable too.
Is there a similiar theorem for functions in $H_0^1(0,1)$?