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 $\varphi$.

Continuity of $u$ at zero supplies finitely many Schwartz seminorms and a constant $C$ bounding $|u[\varphi ]|$. Increasing their two indices to common maxima $N,k$ givesThus $u$ is bounded for the $X_{N,k}$ norm. Since $\mathcal{S}$ is dense in the Banach space $X_{N,k}$, it has a unique continuous extension $\tilde{u}\in X_{N,k}^{\prime }$.

Take the extension from part (a), and choose $M>N+n+1$. The positive functionbelongs to $X_{N,k}$. Positivity extends from nonnegative Schwartz functions to nonnegative elements obtained as $X_{N,k}$ limits. Sincepositivity givesTaking $K=\tilde{v}[w]$ proves that every positive distribution has order zero.