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 ω satisfying the mild condition: 2ω(t) ≤ ω(Ht) + H for some H > 1 and for all t ≥ 0, the space \(\mathcal {S}_\omega \) in the sense of Bjorck is also nuclear.