Let $K$ be a compact subset of $\mathbb{C}$. By definition, one has $$\mathcal{O}'(K) = \left( \varinjlim_{U\supset K} \mathcal{O}(U) \right)',$$ where $U$ are open neighborhoods of $K$. My question is : Do we have $$\mathcal{O}'(K) \simeq \varprojlim_{U\supset K} \mathcal{O}'(U) ?$$ (Here, we are considering strong duals.) These things are in general not true for Fréchet spaces but since we work with very particular and nice spaces, I wonder if it could be true.
Thanks for any help.