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Let $E$ be a locally convex Hausdorff space satisfying the convex compactproperty and let $(T_x)_{x \in \mathbb{R}^d}$ be a locally equicontinuous$C_0$-group of linear continuous operators on $E$. In this article, we showthat if $E$ is quasinormable, then the space of smooth vectors in $E$associated to $(T_x)_{x \in \mathbb{R}^d}$ is also quasinormable. Inparticular, we obtain that the space of smooth vectors associated to a$C_0$-group on a Banach space is always quasinormable. As an application, weshow that the translation-invariant Fr\'echet spaces of smooth functions oftype $\mathcal{D}_E$ [8] are quasinormable, thereby settling the question posedin [8, Remark 7]. Furthermore, we show that $\mathcal{D}_E$ is not Montel if$E$ is a solid translation-invariant Banach space of distributions [10]. Thisanswers the question posed in [8, Remark 6] for the class of solidtranslation-invariant Banach spaces of distributions.

Quasinormable $C_0$-groups and translation-invariant Fr\'echet spaces of type $\mathcal{D}_E$