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?