Extreme confusion with the product of distributions Let $l_1$ and $l_2$ be two distributions in disjoint variables $x_1, ..., x_n$ and $y_1, ..., y_m$. Then it is said to be possible to define a product distribution.
However, I am fundamentally confused. Distributions are in fact linear functionals on the space of smooth and compactly supported functions. Then, how does the 'product' of linear functionals again become a linear functional?
Especially, what can be a definition of $\delta(x_1)\delta(x_2)$ such that it is equal to $\delta(x_1, x_2)$? I am just stuck......

If you want to multiply $\delta(x_1)$ and $\delta(x_2)$ you first need to make them into functions acting on the space space, so you multiply $\delta(x_1)Id(x_2)$ and $Id(x_1)\delta(x_2)$, where $Id$ is just the identity map. With that interpretation you get $\delta(x_1)\delta(x_1)=\delta(x_1, x_2)$.
