About the nuclearity of ${mathcal S}_{(M_p)}$ and ${mathcal S}_{omega}$

We use an isomorphism established by Langenbruch between some sequence spaces and weighted spaces of generalized functions to give sufficient conditions for the (Beurling type) space ${mathcal S}_{(M_p)}$ to be nuclear. As a consequence, we obtain that for a weight function $omega$ satisfying the mild condition: $2omega(t)leq omega(Ht)+H$ for some $H>1$ and for all $tgeq0$, the space ${mathcal S}_omega$ in the sense of Björck is also nuclear