Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 23H / b / Solution

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.