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In the structure theory of Fréchet spaces, especially results around splitting short exact sequences, the properties (DN) and (Ω) play a major rôle. There are many variants, but they are phrased in terms of an increasing sequence $\left(||\cdot||_k\right)_{k=1,2,\ldots}$ of seminorms. In particular, no mention is made of completeness.

It is conceivable that these properties may have been studied for more general locally convex spaces. One might even imagine that variants could be defined where one has a larger system of seminorms. Has this ever been addressed in the literature? Or perhaps variants that take into account the weakening of the assumptions on the tvs?

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  • $\begingroup$ There are splitting results for so-called PLB-spaces (countable projective limits of countable inductive limits of Banach spaces) by Bonet and Domanski where generalizations of DN for those spaces (called "dual interpolation estimates") play an important role. Look at the article ''The splitting of short exact sequences of PLS-spaces and smooth dependence of solutions of linear partial differential equations'', Advances in Mathematics 217, no. 2, (2008) 561-585 $\endgroup$
    – Jochen Wengenroth
    Commented Sep 16, 2017 at 14:27
  • $\begingroup$ What do you want to do with such generalizations? If you are looking for splitting results: It is known e.g. that every short exact sequence $0\to E\to F\to G\to 0$ splits if $E$ is a quotient of the spaces of distribution $\scr D'$, $G$ is a subspace of $\scr D'$ and $F$ is a PLS-space. $\endgroup$
    – Jochen Wengenroth
    Commented Sep 16, 2017 at 14:33
  • $\begingroup$ @Jochen I was wondering about any extensions of splitting theorems past the Fréchet setting, and curious to know how badly things can go wrong when just taking a naïve approach. But also for its own sake. The test case I have in mind is compactly-supported smooth functions. For those following, here's a pdf of the Bonet-Domanski paper jbonet.webs.upv.es/wp-content/uploads/papers/2008/… - thanks for the reference. $\endgroup$
    – David Roberts ♦
    Commented Sep 17, 2017 at 0:18

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