On the projective description of spaces of ultradifferentiable functions of Roumieu type

We provide a projective description of the space $mathcal{E}^{{mathfrak{M}}}(Omega)$ of ultradifferentiable functions of Roumieu type, where $Omega$ is an arbitrary open set in $mathbb{R}^d$ and $mathfrak{M}$ is a weight matrix satisfying the analogue of Komatsus condition $(M.2)$. In particular, we obtain in a unified way projective descriptions of ultradifferentiable classes defined via a single weight sequence (Denjoy-Carleman approach) and via a weight function (Braun-Meise-Taylor approach) under considerably weaker assumptions than in earli