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Let be $\Omega=]-1,1[\times ]-1,1[$. How I will be able to show Whats is the smaller space where $\delta_0$ (delta Dirac) belong?. $\delta_0$ is defined than $\left<{\delta_0,\phi}\right> = \phi(0)$ $\forall\phi\in C^{\infty}_0(\Omega)$

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    If you don't provide context, it is impossible to know what "smaller" could mean.2012-10-22
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    @Martin: 'the smaller' intends to mean 'the smallest'. But some context would be good, yes.2012-10-22
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    @Berci: indeed. Still, we don't know whether it's the smallest vector space, Banach space, Hilbert space. And even if we fix one of those categories, it might not be obvious what "smaller" means. My answer would be that the smallest space that contains $\delta_0$ is $\mathbb C\delta_0$, but I doubt this is what the asker wants.2012-10-23
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    It looks to me like you are trying to define a distribution (or even a measure) $\delta_0$ on $\Omega$, and asking what the support of the distribution is. I have one concern about your definition, though. What does "$\phi(0)$" mean, when there is no point $0$ in $\Omega$?2012-10-23
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    My teacher wrote this excercise, ... when the vector space is Banach for example2012-10-23
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    I asked about space and yes is Banach Space2012-10-23
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    $\phi(0)$ is a $\phi$ evaluated in the point zero2012-10-23
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    @Juan: Notice that $\Omega$ does not contain the point $0$. If $\phi\in C_0^\infty(\Omega)$, then it follows that $0$ does not lie in the support of $\phi$. Unless I am misinterpreting your question, the only sensible way to define $\phi$ at $0$ is by setting $\phi(0) = 0$. But then $\langle \delta_0,\phi\rangle = 0$ for each $\phi$, i.e., $\delta_0 = 0$.2012-10-23
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    @froggie understand mmm, I thinking ...2012-10-24
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    @froggie then Whats is a smaller space that contains a zero function ?2012-10-24
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    I edit a question my teacher tell that wrote with mistake2012-10-24

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