Let $T:C^{1}_{[a,b]} \rightarrow C^{0}_{[a,b]}$ with $a<b$ be the differential operator defined as $Tx=x’$.
The practice exercise asks for the kernel and range of such operator and also a demonstration using the definition of bounded operators.
An operator on normed spaces is said to be bounded when $\|Tx\| \leq c\|x\|$ where $c$ is a real number.
I have already been through examples of continuous functions which show that it is not bounded. However, this one specifically asks to use the definition.
Your contribution would be much appreciated. Feel free to shut this down if you indicate a solution that is available out there.
Best Answer
Take $[a,b]=[0,1],$ let $n\in\mathbb{N}$ be arbitrary. Next, consider the sequence $f_n(t)=t^n.$ Then, $Tf_n(t)=nt^{n-1}.$ If we equip $C^1$ with the infinity norm, then $||Tf_n||_{\infty}=n,$ but $||f_n||_\infty=1.$ So, $$\frac{||Tf_n||_{\infty}}{||f_n||_\infty}=n.$$ By assumption, this is true for any $n$. Can you see why this means there is no $c$ for which $||Tf_n||_\infty\leq c||f_n||_\infty?$