$P$ is a vector space of all polynomial functions on $C[0,1]$
$p'$ is derivative of $p$.
Show that $T : P \to P \quad T(p)= p'$ is closed operator (the graph of T is closed) but not continuous.
Use $|| \; ||_{sup}$ (supremum norm) for polynomials.
On wikepedia it says that differentiation operator is not continuous how can I show it for this problem ?
Best Answer
Take any polynomial $x^{n}$. The sup norm of this is $1$. It's derivative is $nx^{n-1}$, the norm of which is $n$. Hence, the norm of the derivative can be made arbitrarily large. So the map is not bounded.