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 $K$?

  • 2
    $\begingroup$ Related questions: mathoverflow.net/questions/145215/… and mathoverflow.net/questions/105147/… $\endgroup$ – András Bátkai Jun 20 '18 at 11:09
  • $\begingroup$ 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 $\mathbb{R}$. $\endgroup$ – abx Jun 20 '18 at 13:44
  • $\begingroup$ @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? $\endgroup$ – user125821 Jun 20 '18 at 15:08
  • $\begingroup$ I don't. This is why the answer is negative. $\endgroup$ – abx Jun 20 '18 at 16:59
  • $\begingroup$ @abx, sorry I don't understand. What are Lebesgue measures for you? $\endgroup$ – Tomek Kania Jun 20 '18 at 17:41

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.