Smaller space $\delta_0$ 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 $\left<{\delta_0,\phi}\right> = \phi(0)$ $\forall\phi\in C^{\infty}_0(\Omega)$
 A: It is still unclear what you mean by "the smaller" space $\delta_0$ belongs to. I think there are two possible good answers to this question.
Answer 1: Right now you have defined $\delta_0$ to be a linear map $C^\infty_0(\Omega)\to \mathbb{R}$ (or $\mathbb{C}$, depending on whether you are working over the real or complex numbers; I'll work over $\mathbb{R}$). That is, $\delta_0$ is an element of the dual space $C^\infty_0(\Omega)^*$. However, notice that the definition of $\langle\delta_0, \phi\rangle$ does not depend on any derivatives of $\phi$, it only cares what the value of $\phi$ is at $0$. In fact, $\delta_0$ defines a continuous linear map $C_0(\Omega)\to \mathbb{R}$. Thus you can think of $\delta_0$ as an element of the dual space $C_0(\Omega)^*$, which is smaller than $C_0^\infty(\Omega)^*$.
Answer 2: Another thing you may be asking is, what is the support of $\delta_0$? You should think of the support of $\delta_0$ as the smallest closed subset of $\Omega$ on which $\delta_0$ takes nonzero values. More precisely, the complement of the support of $\delta_0$ is the largest open set $U\subseteq\Omega$ such that $\langle \delta_0,\phi\rangle = 0$ for all $\phi\in C_0^\infty(U)$. In this problem, notice $\langle \delta_0, \phi\rangle = 0$ for every $\phi\in C_0^\infty(\Omega\smallsetminus\{0\})$, so the support of $\delta_0$ is the point $\{0\}$.
My guess is answer 1 is what you are looking for, but hopefully one of these helped.
