Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 23H (Analysis of Functions)

The Schwartz space $\mathcal{S}({\mathbb{R}}^n)$ consists of smooth functions for which every seminormis finite. Convergence means convergence in every such seminorm. A tempered distribution is a continuous linear functional on this Fréchet space; $u_j\to u$ in ${\mathcal{S}}^{\prime }$ means $u_j[\varphi ]\to u[\varphi ]$ for every test function $\varphi$.