I have been trying to understand the following section of a paper "Revêtements du demi-plan de Drinfeld et correspondance de Langlands p-adique" by Gabriel Dospinescu and Arthur-César Le Bras:

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La dualité de Serre pour les variétés Stein [

13] montre que ce complexe est dual du complexe des sections globales du complexe de de Rham de $X$, tordu par $\Omega^d(X)$. Cela permet de montrer [47, th. 4.11] que si $X$ est pure de dimension $d$, alors pour tout $k$ on a des isomorphismes canoniques $$H^k_\text{dR}(X) \simeq H^{2d - k}_\text{dR,c}(X)^*\quad\text{et}\quad H^k_\text{dR,c}(X) = H^{2d - k}_\text{dR}(X)^*,$$ les duaux étant topologiques (comme toujours dans cet article). La preuve de [46, cor.3.2] montre que pour tout $k$ l'espace vectoril topologique $H^k_\text{dR}(X)$ est isomorphe à la limite inverse d'une suite $(V_n)_n$ d'espaces de dimension finie sur $K$. En particulier $H^k_\text{dR}(X)$ est un Fréchet réflexif et son dual topologique $H^{2d - k}_\text{dR,c}(X)$ est la limite inductive des $V_n^*$.On en déduit que $H^k_\text{dR}(X)$ est aussi le dual algébrique de $H^{2d - k}_\text{dR,c}(X)$.Puisque $\Omega$ est un espace Stein^{(23)}, il en est de même de $\Sigma_0$ et puisue $\Sigma_n$ est un revêtement étale fini de $\Sigma_0$, on obtient la …

(the full version of a preprint of their paper can be found here: https://arxiv.org/abs/1509.00606). Above, $K$ is a field of characteristic zero, complete with respect to a discrete valuation and $X$ is a smooth, rigid analytic $K$-space. *I cannot prove the sentence underlined in green.* My suspicion is that this is a general result: if $Y$ is the inverse limit¹ of finite dimensional $K$-vector spaces then its continuous dual $Y'$ and its algebraic dual $Y^*$ coincide. So far, I believe that I can prove that $(Y')' \cong (Y')^*$ as vector spaces, and have tried proceeding using reflexivity. However, I can't get to the final result and I'm now concerned that I am missing something silly. Any help would be hugely appreciated!

¹over the non-negative integers. We may also need to impose a density condition, i.e. that if $Y= \varprojlim_{n \geq 0} Y_n$ then the projection maps $Y \rightarrow Y_n$ have dense image for all $n \geq 0$. This allows us to apply results like [1, Theorem 11.6.1].

- Perez-Garcia C, Schikhof WH. Locally Convex Spaces over Non-Archimedean Valued Fields. Cambridge University Press; 2010.

from an inductive limitinto $K$. However, I can't see how they establish that $Y' = Y^*$ because there we are looking at mapsfrom a projective limit(which is equipped with an initial topology) and so its harder to use the first fact you mentioned about finite dimensional vector spaces. I'm probably missing something obvious. It wouldn't be the first time! I'll think through what you have written again :) $\endgroup$