Timeline for The dual space of the Dirac measures on an Abelian group
Current License: CC BY-SA 3.0
10 events
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| Apr 13, 2017 at 12:57 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Jan 23, 2014 at 13:39 | comment | added | Liviu Nicolaescu | There is a result valid for any locally compact group: the convolution of any Borel regular measure with the Dirac measure at the origin is the original measure. If you unravel the definition of convolution of measure you deduce that any measure is a "superposition" of $\delta$'s or, as you put it, an "integral" combination of $\delta$'s | |
| Jan 23, 2014 at 8:09 | comment | added | Juan Bermejo Vega | At the very beginning I was actually hoping that merely taking the space of continuous (or, perhaps, uniformly continuous) functions in $L^2(G)$ would be enough. (Let's call such spaces $C(G)$ and $C_u(G)$.) Given that, I was hoping that taking $\mathcal{R}$ to be $C(G)\cap C(\widehat{G})$ or $C_u(G)\cap C_u(\widehat{G})$ would give you a dual $\mathcal{R}^\times$ containing all Dirac measures of $G$ and $\widehat{G}$ and "only them". I could not work this out, though; in particular, I do not if $\mathcal{R}$ would be dense. | |
| Jan 23, 2014 at 8:00 | comment | added | Juan Bermejo Vega | The space you mention is what I called $C_c(G)$. I know that there is a version of the [Riesz–Markov–Kakutani] theorem showing that $C_c(G)^\times$ is precisely the space of regular Borel measures on $G$. However, I do not know how to apply this result: are such measures "integral combinations" of Dirac measures? | |
| Jan 23, 2014 at 7:52 | comment | added | Juan Bermejo Vega | I am just interested on constructing the rigging, so, I would take the simplest topology such that the inclusions in the containment $\mathrm{R}⊂L^2(G)⊂\mathrm{R}^\times ×$ are continuous. Would that be the subspace topology in $L^2(G)$? | |
| Jan 20, 2014 at 15:11 | answer | added | alpha | timeline score: 1 | |
| Jan 20, 2014 at 14:35 | comment | added | Liviu Nicolaescu | When you talk of continuous linear functionals on $\mathscr{R}$ you tacitly assume that $\mathscr{R}$ is a topological vector space. What topology are you thinking of? (The space of continuous compactly supported functions will do the trick.) | |
| Jan 20, 2014 at 13:45 | history | edited | Juan Bermejo Vega | CC BY-SA 3.0 |
deleted 115 characters in body; edited title
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| Jan 20, 2014 at 13:23 | history | edited | Juan Bermejo Vega | CC BY-SA 3.0 |
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| Jan 20, 2014 at 13:16 | history | asked | Juan Bermejo Vega | CC BY-SA 3.0 |