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arXiv:2403.08296v2 Announce Type: replace
Abstract: We prove that the spaces of smooth and ultradifferentiable vectors associated with a representation of a real Lie group on a Fr\'{e}chet space $E$ are quasinormable if $E$ is so. A similar result is shown to hold for the linear topological invariant $(\Omega)$. In the ultradifferentiable case, our results particularly apply to spaces of Gevrey vectors of Beurling type. As an application, we study the quasinormability and the property $(\Omega)$ for a broad class of Fr\'{e}chet spaces of smooth and ultradifferentiable functions on Lie groups globally defined via families of weight functions.
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