Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 9 / 4 / a / Solution

Weak convergence of distributions means $⟨T_n,\phi ⟩\to ⟨T,\phi ⟩$ for every test function $\phi \in C_c^{\infty }(\Omega )$. The distributional derivative is defined byIt is linear and continuous: differentiation preserves compact support and bounds each test-function seminorm by one of higher derivative order, so composing it with the continuous functional $T$ again gives a distribution. Finally, for every fixed test function,Thus distributional differentiation preserves the stated convergence.