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Setting:

Let $X'$ be a dense subset of an infinite-dimensional Fréchet space $X$ and suppose that $(X_n')_{n \in \mathbb{N}}$ is a nested sequence of non-empty subsets of $X'$ satisfying $$ \bigcup_{n \in \mathbb{N}} X_n'=X - N , $$ where $N$ is a nowhere dense subset of $X$.

Question:

Are there reasonable conditions on $\{X_n'\}$, such that, for every $n \in \mathbb{N}$, there are topologies $\{\tau_n\}$ on $X$ satisfying:

  • $1_X:(X,\tau_{n+1})\rightarrow (X,\tau_{n})$ is continuous but the inverse map is not,
  • $\varprojlim_n (X,\tau_n) = X$
  • $\bigcup_{i=1}^n X_i'$ is dense in $(X,\tau_n)$,
  • $\tau_0$ is non-trivial, in the sense that: $\tau_0\neq \{\emptyset,X\}$.
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If $X$ is a proper Fréchet space (i.e., not isomorphic to a Banach space) you can choose an increasing sequence of semi-norms $p_n$ giving the Fréchet space topology of $X$ such that the $p_{n}$-topology $\tau_n$ is strictly finer than $\tau_{n-1}$. These semi-normed topologies satisfy you requirements (although non-trivial is not very specific).

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  • $\begingroup$ Do you have a reference? For example, it's not clear to me how this would make $X_n'$ dense in the $p_n$-topology? $\endgroup$
    – ABIM
    Commented Jun 24, 2020 at 11:21
  • $\begingroup$ If it is dense in $X$ it remains dense with respect to the coarser topologies $\tau_n$. $\endgroup$
    – Jochen Wengenroth
    Commented Jun 24, 2020 at 11:36
  • $\begingroup$ But $\bigcup_n X_n'$ is dense in $X$ (and therefore the coarser topologies) but why is $X_n'$ dense (on it's own) in $\tau_n$ (or $\bigcup_{i=1}^n X_i'$ dense in $\tau_n$)? $\endgroup$
    – ABIM
    Commented Jun 24, 2020 at 12:56
  • $\begingroup$ I described the standard way to make a Fréchet space the projective Limit of (semi-) normed spaces. In general, the sets $X_n'$ won't be dense. For example, each $X_n'$ could be a finite dimensional subspace, if the $p_n$ are proper norms (and not only semi-norms) then they are closed. What do you want to do with construction of the $\tau_n$? $\endgroup$
    – Jochen Wengenroth
    Commented Jun 24, 2020 at 14:39
  • $\begingroup$ Well, for me $X=C(\mathbb{R},\mathbb{R})$, $\cup X_n'$ is a set of continuous functions and I want to describe the "approximation-theoretic impact" of adding $X_{n+1}'-X_n'$. In my setting, $X_n'$ are affine subspaces (or "linear manifolds" as some authors say). I guess, it would be more appropriate if $X$ is instead required to be a (strict) LB/LF-space? $\endgroup$
    – ABIM
    Commented Jun 24, 2020 at 16:39

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