Solution
= Solution
Take the extension from part (a), and choose $M>N+n+1$. The positive function
$$
w(x)=(1+|x|^2)^{-M/2}
$$
belongs to $X_{N,k}$. Positivity extends from nonnegative Schwartz functions to nonnegative elements obtained as $X_{N,k}$ limits. Since
$$
-\|\phi\|_{M,0}w\leq\phi\leq\|\phi\|_{M,0}w,
$$
positivity gives
$$
|v[\phi]|\leq \widetilde v[w]\,\|\phi\|_{M,0}.
$$
Taking $K=\widetilde v[w]$ proves that every <positive distribution> has order zero.