We give a new proof of the completeness of the space $\mathcal{ O}_C$ by applying a criterion of compact regularity for the isomorphic sequence space $\lim_{ k\rightarrow} (s\hat \otimes (\ell^\infty)_{ -k})$. Along the way we show that the strong dual of any quasinormable Fr\'echet space is a compactly regular $\mathcal{ LB}$-space. Finally, we prove that $\lim_{ k\rightarrow}(E_k\hat \otimes_\iota F) = (\lim_{ k\rightarrow} E_k) \hat \otimes_\iota F$ if the inductive limit $\lim_{ k \rightarrow}(E_k \hat \otimes_\iota F)$ is compactly regular.
Categories: math.FA