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: $2\omega (t)\le \omega (Ht)+H$ for some $H>1$ and for all $t\ge 0$, the space ${\mathcal{S}}_{\omega }$ in the sense of Bj"orck is also nuclear.