Whilst reading the article about restrictions of distributions (generalized functions) on Wikipedia (here)
I had trouble understanding the example of a distribution defined on the subset $V = (0,2) \subset \mathbb{R}$ that admits no extension to the space of Distributions on $U = \mathbb{R}$.
The example I am referring to is the distribution
\begin{equation} S(x) = \sum_{n = 1}^\infty n \delta \left(x - \frac{1}{n}\right) \end{equation}
Now, my question is how does this example act on test functions ? If I take a smooth function $\psi$ whith supp $\psi \subset V$, how do I apply $S$ to it ? I understand that $S$ is a modification of the Dirac delta distribution
\begin{equation} \langle \delta , \psi \rangle = \psi (0) \end{equation}
Below is my guess :
\begin{equation} \langle S , \psi \rangle = \sum^{\infty}_{n = 1} n \psi (\frac{1}{n}) \end{equation}
Is that correct ? If yes, I am still not sure how this is well defined, given only the restriction that $\psi$ is zero outside $(0,2)$. Tanks a lot for your help!