Let $k$ be a field and $\{V_i\}_{i \in I}$ a filtered projective system of $k$-spaces with transition maps $f_{ji}: V_j \rightarrow V_i$ for $i \leq j$ (for my purposes we may assume the index set is $\mathbb{N}$). For each $i$, let $V_i^{**}$ be the double dual of $V_i$: here this is simply the algebraic dual, $Hom_k(Hom_k(V_i, k), k)$. The $V_i^{**}$ also form an projective system.

If $\varprojlim V_i^{**}$ is finite-dimensional over $k$, do we have $\varprojlim V_i \simeq \varprojlim V_i^{**}$? (Here there is no assumption that the $V_i$ themselves are finite-dimensional.)

Since $V_i \hookrightarrow V_i^{**}$ canonically for each $i$ and projective limit is left-exact, it follows that $\varprojlim V_i \hookrightarrow \varprojlim V_i^{**}$. Therefore the hypothesis implies that $\varprojlim V_i$ is also finite-dimensional, and isomorphic to its own double dual $(\varprojlim V_i)^{**}$. It would suffice to show that there is an injection $\varinjlim V_i^* \hookrightarrow (\varprojlim V_i)^*$, as this would dualize to a surjection $(\varprojlim V_i)^{**} \twoheadrightarrow (\varinjlim V_i^*)^* = \varprojlim V_i^{**}$ and the proof would be complete by comparing dimensions. There is a natural map $\varinjlim V_i^* \rightarrow (\varprojlim V_i)^*$ , induced by the duals of the structural maps $\pi_i: \varprojlim V_i \rightarrow V_i$, but even with the assumption that both source and target are finite-dimensional I can't seem to show that this map is injective (nor can I think of a counterexample).