A sequence space representation of L. Schwartz' space \({{\mathcal{O}_{C}}}\) (Q420965)

In the late 1970's and early 1980's Valdivia and Vogt independently obtained sequence space representations of many spaces of \(C^\infty\)-functions and distributions. Their work solved several open problems about the existence of bases in function spaces and clarified the structure of many function and distribution spaces. \textit{M. Valdivia} in [Math. Z. 177, 463--478 (1981; Zbl 0472.46025)] proved that the space \(O_M\) of Schwartz of slowly increasing functions is isomorphic to the complete projective tensor product \(s' \hat{\otimes}_\pi s\). However, a sequence space representation of its dual \(O'_M\), which is isomorphic to the space \(O_C\) of very slowly increasing functions, was missing. The author applies Valdivia's representation and some results on (duality) of tensor products due to Grothendieck to obtain that \(O_C\) is isomorphic to the complete tensor product \(s' \hat{\otimes}_i s\). The topology \(i\) is the finest locally convex topology on \(E \otimes F\) that makes the canonical bilinear map \(E \times F \rightarrow E \otimes F\) separately continuous. The author also presents an explicit representation of \(s' \hat{\otimes}_i s\) as a complete space of double sequences, that is a countable inductive limit of vector valued Fréchet sequence spaces.