Timeline for Dual space of continuous functions
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Feb 23, 2015 at 16:21 | comment | added | user11178 | Dunford Schwarz | |
| Dec 18, 2011 at 10:26 | comment | added | Mariarty | The only thing I want to figure out, which reference I should give to the fact of the uniqueness of the extension? Because, as I know, the definition of Stone-Cech compactification is based on mapping $X$ to $[0,1]^C$, where $C$ is the set of all continuous functions from $X$ into $[0,1]$. After that question can be closed. | |
| Dec 16, 2011 at 23:14 | comment | added | Bill Johnson | Since the OP is satisfied with the answers given in the comments, I vote to close. | |
| Dec 16, 2011 at 11:22 | comment | added | Mariarty | Thank you very much! This uniqueness of the extension is exactly what I needed. And now I also agree that there isn't "explicit" description. | |
| Dec 16, 2011 at 9:49 | comment | added | Yemon Choi | It follows from the universal properties of $\beta{\mathbb R}$ that any continuous bounded function on ${\mathbb R}$ extends uniquely to a continuous function on $\beta{\mathbb R}$, so that is how you define $f(\omega)$. One gets the representing measure $\mu$ by applying the Riesz representation theorem for $C(\beta{\mathbb R})$. If you are asking for an "explicit" description of the representing measure, then I think the question that Matthew Daws links to suggests that this is unlikely | |
| Dec 16, 2011 at 8:38 | comment | added | Mariarty | Thank you. Could you please clarify the form of the functional? Will it be the same? $$ x^*f=\int\limits_{\beta\mathbb R}f(\omega)\mu(d\omega) $$ And how we understand $f(\omega)$ when $\omega\in\beta\mathbb R\setminus \mathbb R$? By the "beautiful" I meant, for example in our case, that functional consists of two parts: integral on $\mathbb R$ and on $\beta\mathbb R\setminus \mathbb R$ (because I don't know about measurability in $\beta\mathbb R$) | |
| Dec 16, 2011 at 7:50 | comment | added | Matthew Daws | A related question: mathoverflow.net/questions/44183/… In particular, Grey Kuperberg's answer makes me skeptical you will get a "good" answer... | |
| Dec 16, 2011 at 5:14 | comment | added | Yemon Choi | The dual of $C_b({\mathbb R})$ is the space of Radon measures on the Stone-Cech compactification of ${\mathbb R}$ -- I don't know if that is beautiful enough. What kinds of result were you hoping to find? | |
| Dec 16, 2011 at 4:52 | history | edited | Mariarty | CC BY-SA 3.0 |
improved formatting
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| Dec 16, 2011 at 4:45 | history | asked | Mariarty | CC BY-SA 3.0 |