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This is a variant of this question.

Definitions/Facts


  • $Ball_1(X)$ denotes the unit ball (about $0$) in a normed vector space $X$.
  • MixTop of triples of pairs $(X,\tau)$ of normed vector spaces $X$ equipped with a LC (locally convex) topology $\tau$ on $X$ making $Ball_1(X)$ complete and bounded together with short linear maps whose restriction to the unit ball is $\tau$-continuous.
  • MixTop contains the category $Ban_1$ of Banach spaces and continuous short linear maps and similarly it is complete, see this article.

Question


Let $(X_n,\pi_n^{m})$ be a countable projective system in MixTop,thus $\projlim X_n$ is well-defined therein. Suppose further that, a strict variant of the Mittag-Leffer condition holds true: $$ \pi_m^n(Ball_1(X_n)) \subseteq \overline{\pi_k^n(Ball_1(X_k))} \qquad \forall k \geq n, $$ and that $U\subseteq \projlim X_n$ such that: $$ \overline{\pi_n(U)}=X_n, $$ then is $U$ dense in $\projlim X_n$ itself?

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    $\begingroup$ You will find a more comprehensive treatment inthe book „Saks spaces and applications to functional analysis“ by the same author. The appropriate assumptions for a theorem of this type would be a projective limit of Banach spaces so that the images of the unit balls are dense in the balls of the image space. $\endgroup$
    – user131781
    Jan 13, 2020 at 14:58
  • $\begingroup$ I'm reading the suggested monograph, however I can't find the said theorem. Is is known? $\endgroup$
    – ABIM
    Jan 13, 2020 at 15:16
  • $\begingroup$ Not to my knowledge. I guess that it follows from the Bourbaki version, resp. its proof but I have no access at the moment. Best to work on a proof yourself. $\endgroup$
    – user131781
    Jan 13, 2020 at 15:56

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