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

Abstract

We explicitly prove the compact regularity of the $\mathrm{LF}$-space of double sequences ${\lim }_{k\to }\left(s\hat{\otimes }{\left({\ell }^p\right)}_k\right)\cong {\lim }_{k\to }\left(s\hat{\otimes }{\left(c_0\right)}_{-k}\right)$, $1\le p\le \infty$. As a consequence, we obtain that the spaces of slowly and uniformly slowly increasing $C^{\infty }$-functions $O_M$ and $O_C$, respectively, are ultrabornological and complete. Furthermore, we prove that ${\lim }_{k\to }\left(E_k{\hat{\otimes }}_{\iota }F\right)=\left({\lim }_{k\to }{E}_k\right){\hat{\otimes }}_{\iota }F$ if the inductive limit ${\lim }_{k\to }\left(E_k{\hat{\otimes }}_{\iota }F\right)$ is compactly regular.