I've read somewhere that the kernel of a linear map is closed iff the map is bounded.
Consider the derivative operator $D: \mathcal C^1([0,1],\mathbb C) \to \mathcal C([0,1],\mathbb C)$, i.e. $Df = f'$.
Since this map is not bounded, the kernel must not be closed.
I'm a bit confused. I thought the kernel of this map was the constant functions. I don't see why this kernel wouldn't be closed.