A splitting theorem for subspaces and quotients of \({\mathcal D'}\) (Q2777729)

The author improves a recent splitting theorem of \textit{P. Domaánski} and \textit{D. Vogt} [Stud. Math. 140, 57-77 (2000; Zbl 0973.46067)]. In this article, the authors had derived a complete splitting theory for the space \({\mathcal D}'\) of distributions (on some open subset of \(\mathbb{R}^d)\).NEWLINENEWLINENEWLINEAn LS- (resp., LN-) space is an inductive limit of a sequence of Banach spaces with compact (resp., nuclear) linking maps. A PLS- (resp., PLN-) space is a reduced countable projective limit of LS- (resp., LN-) spaces. An (algebraically and topologically) exact sequence of PLS-spaces NEWLINE\[NEWLINE0\to F @>j>> X @>q>> G\to 0\tag{*}NEWLINE\]NEWLINE splits if \(j(F)\) is complemented in \(X\) or, equivalently, if \(q\) has a continuous linear right inverse. The splitting of all such sequences (*), where \(F\) and \(G\) are fixed PLS-spaces and \(X\) is also a PLS-space, is denoted by \(\text{Ext}^1_{\text{PLS}} (G,F)=0\).NEWLINENEWLINENEWLINEDomańsski and Vogt had proved that an ultrabornological PLN-space \(F\) is isomorphic to a quotient of \({\mathcal D}'\) if and only if \(\text{Ext}^1_{\text{PLS}} ({\mathcal D}',F) =0\), and that a PLN-space \(F\) is isomorphic to a subspace of \({\mathcal D}'\) if and only if \(\text{Ext}^1_{\text{PLS}}(F,{\mathcal D}')=0\). The main result of the present article shows that for PLS-spaces \(E\) and \(F\) such that \(E\) is isomorphic to a subspace of \({\mathcal D}'\) and \(F\) is isomorphic to a quotient of \({\mathcal D}'\) the relation \(\text{Ext}^1_{\text{PLS}}(E,F)=0\) holds. The main ingredients of the proof are a construction which can be carried out in any quasi-abelian category and, from the article of Domański and Vogt, the existence of a short exact sequence \(0\to {\mathcal D}'\to {\mathcal D}'\to F\to 0\) whenever \(F\) is isomorphic to a quotient of \({\mathcal D}'\).