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

TL;DR

This paper provides a new proof of the completeness of the space O_C using a criterion of compact regularity, and explores duality and tensor product properties of certain topological vector spaces.

Contribution

It introduces a novel proof technique for the completeness of O_C and investigates duality and tensor product regularity in quasinormable Fre9chet spaces.

Findings

Established the completeness of O_C via compact regularity.

Showed the strong dual of any quasinormable Fre9chet space is a compactly regular LB-space.

Proved tensor product limit interchange under compact regularity conditions.

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 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.