Timeline for Dual of $C(X)$ with the compact open topology
Current License: CC BY-SA 4.0
11 events
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| Jun 20, 2018 at 18:34 | comment | added | abx | @Tomek Kania: The restriction of the Lebesgue measure on $\Bbb{R}$, what else? | |
| Jun 20, 2018 at 18:30 | comment | added | Pietro Majer | Take for instance the discrete space $X:=\mathbb{N}$. Then $C(X)$ is the projective limit $\displaystyle{\lim_\leftarrow } $ of the embeddings $\mathbb{R}^n\to\mathbb{R}^{n+1}$, that is the Fréchet space $\mathbb{R}^\mathbb{N}$ with the product topology, and this is also the projective limit of $C(K)^*$ for $K\Subset\mathbb{N}$. On the other side, $C(X)^*$ is the space $c_c$ of compactly supported sequences, that is the inductive limit $\displaystyle{\lim_\rightarrow } $ of the $\mathbb{R}^n\to\mathbb{R}^{n+1}$, topologized by all seminorms on it, a complete, non metrizable LCTVS. | |
| Jun 20, 2018 at 17:41 | comment | added | Tomasz Kania | @abx, sorry I don't understand. What are Lebesgue measures for you? | |
| Jun 20, 2018 at 16:59 | comment | added | abx | I don't. This is why the answer is negative. | |
| Jun 20, 2018 at 15:08 | comment | added | user125821 | @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? | |
| Jun 20, 2018 at 13:44 | comment | added | abx | 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 $\mathbb{R}$. | |
| Jun 20, 2018 at 11:09 | comment | added | András Bátkai | Related questions: mathoverflow.net/questions/145215/… and mathoverflow.net/questions/105147/… | |
| Jun 20, 2018 at 10:34 | history | edited | user125821 | CC BY-SA 4.0 |
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| Jun 20, 2018 at 10:25 | history | edited | user125821 | CC BY-SA 4.0 |
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| Jun 20, 2018 at 10:12 | review | First posts | |||
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| Jun 20, 2018 at 10:09 | history | asked | user125821 | CC BY-SA 4.0 |