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

Abstract

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 ${\lim }_{k\to }(s\hat{\otimes }({\ell }^{\infty }{)}_{-k})$. Along the way we show that the strong dual of any quasinormable Fréchet space is a compactly regular $\mathcal{LB}$-space. Finally, we prove that ${\lim }_{k\to }(E_k{\hat{\otimes }}_{\hat{I}}^{1}F)=({\lim }_{k\to }{E}_k){\hat{\otimes }}_{\hat{I}}^{1}F$ if the inductive limit ${\lim }_{k\to }(E_k{\hat{\otimes }}_{\hat{I}}^{1}F)$ is compactly regular.

11 pages, final version