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We use an isomorphism established by Langenbruch between some sequence spacesand weighted spaces of generalized functions to give sufficient conditions forthe (Beurling type) space ${\mathcal S}_{(M_p)}$ to be nuclear. As aconsequence, we obtain that for a weight function $\omega$ satisfying the mildcondition: $2\omega(t)\leq \omega(Ht)+H$ for some $H>1$ and for all $t\geq0$,the space ${\mathcal S}_\omega$ in the sense of Bj\"orck is also nuclear.

About the nuclearity of ${\mathcal S}_{(M_{p})}$ and ${\mathcal S}_{\omega}$