On the completeness of the space ${\mathcal{O}}_C$

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 ${\lim }_{k\to }(s\widehat{\otimes }({\mathrm{\ell }}^{\mathrm{\infty }}{)}_{-k})$. Along the way we show that the strong dual of any quasinormable Fréchet space is a compactly regular $\mathcal{L}\mathcal{B}$-space. Finally, we prove that ${\lim }_{k\to }(E_k{\widehat{\otimes }}_{\iota }F)=({\lim }_{k\to }{E}_k){\widehat{\otimes }}_{\iota }F$ if the inductive limit ${\lim }_{k\to }(E_k{\widehat{\otimes }}_{\iota }F)$ is compactly regular.