# Why is the derivative operator never bounded?

Consider $$X= C^\infty ([0,1],\mathbb{R} )$$ and the operator $$D: (X,\|\cdot\| ) \to (X,\|\cdot\|).$$ given by $$D(x)=x'.$$ (derivation operator).

Why is unbounded, independently of the choice of norm? I can prove this for cases where I know the norm, but why does this hold in the general case?

• limited = bounded? – daw Oct 27 at 20:09

Because the exponential functions $$e^{cx}$$ with $$c\in\mathbb{R}$$ exist in $$C^\infty([0,1],\mathbb{R})$$: $$\|D(e^{cx})\|=\|ce^{cx}\|=|c|\cdot\|e^{cx}\|,$$ and there is no constant $$M\geqslant0$$ such that this norm is $$\leqslant M\cdot\|e^{cx}\|$$ for all $$c \in \mathbb{R}$$.