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: $2\omega(t)\leq \omega(Ht)+H$ for some $H>1$ and for all $t\geq0$, the space ${\mathcal S}_\omega$ in the sense of Bj\"orck is also nuclear.