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arXiv · 2408.11944

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

Michael Kunzinger·Norbert Ortner
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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 $\lim_{k\rightarrow} (s\hat \otimes (\ell^\infty)_{-k})$. 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 $\lim_{k\rightarrow}(E_k\hat \otimes_ιF) = (\lim_{k\rightarrow} E_k) \hat \otimes_ιF$ if the inductive limit $\lim_{k \rightarrow}(E_k \hat \otimes_ιF)$ is compactly regular.

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Michael Kunzinger, Norbert Ortner. 2025-07-01. On the completeness of the space $\mathcal{O}_C$. https://arxiv.org/abs/2408.11944

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