# Convergence to the differential operator

This is not an accurate question. I am not so sure about unbounded operators on a Hilbert space.

Let $$D$$ be the differential operator on $$L^2(0,1)$$. Well, to somewhat, we can extend $$D$$ to a normal operator, still denoted by $$D$$. So, it has a spectral representation, $$D:=\int \lambda dE_\lambda$$. My question is, are there any sequences of bounded linear operators on $$L^2(0,1)$$ converging to $$D$$ in some topology? and what is this topology?