# Dual of $C(X)$ with the compact open topology

Let $X$ be a completely regular space and let $C_k(X)$ be the space of all continuous functions with the compact-open topology. If $X$ is completely metrizable, is the strong dual $C(X)^*$ the strong projective limit of the spaces $C(K)$, where $K$ is a compact subset of $K$?

• Related questions: mathoverflow.net/questions/145215/… and mathoverflow.net/questions/105147/… – András Bátkai Jun 20 '18 at 11:09
• No. By the references mentioned by András Bátkai, $\mathcal{C}(X)^*$ can be identified with the space of (Radon) measures on $X$ with compact support. This space maps by restriction to each $\mathcal{C}(K)^*$, hence to their projective limit, but the latter map is not surjective: the family of Lebesgue measures on compacts of $\mathbb{R}$ does not come from a compactly supported measure on $\mathbb{R}$. – abx Jun 20 '18 at 13:44
• @abx, but if you take such a measure $\mu$ (supported on the whole real line), how do you guarantee that $\int f d \mu$ is finite for every (unbounded) continuous function? – user125821 Jun 20 '18 at 15:08
• I don't. This is why the answer is negative. – abx Jun 20 '18 at 16:59
• @abx, sorry I don't understand. What are Lebesgue measures for you? – Tomek Kania Jun 20 '18 at 17:41