Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 327 / 2 / Solution

The space of test functions is $\mathcal{D}(\mathbb{R})=C_c^{\infty }(\mathbb{R})$. A sequence ${\phi }_j$ converges to $\phi$ when its supports eventually lie in a single compact set and every derivative converges uniformly. The distribution space ${\mathcal{D}}^{\prime }(\mathbb{R})$ is its continuous linear dual; weak convergence means $⟨u_j,\phi ⟩\to ⟨u,\phi ⟩$ for every test function. We use $⟨u^{\prime },\phi ⟩=-⟨u,{\phi }^{\prime }⟩$ and $⟨au,\phi ⟩=⟨u,a\phi ⟩$ for multiplication of a distribution by a smooth function.