Define $T:X\rightarrow L^{p}[0,1]$ where $Tf=f'$. Here the function $f$ is in $C^{\infty}[0,1]$ and $f(0)=0$. How do I show that this $T$ is an injective linear operator that is not bounded? Correct me if I am wrong, the prime here refers to the differential operator.
To show that it is linear is pretty straightforward. But how is it that $T$ is injective? One to one?
Also, how do I show that it is unbounded? Do I come up with a specific function $f$?
Lastly, what about the inverse of $T$? Is it unbounded too?
Just a note: I have read some articles and it seems like letting $f_{k}=sinkx$ will do the job but the norm is infinity norm. Here $p$ is finite.