How would one show that $$T: \mathcal D \to\mathbb R $$ $$\psi\to \newcommand\inner[2]{\langle #1,#2 \rangle}\inner{T}{\psi} = \sum_{n\in\mathbb Z} \psi (n) $$ is a distribution.
I know that $T$ is a distribution in for every compact set $K$, one can find constants $c$ and $j$ such that $\|\inner {T}\psi{}\|\leq c ×sup_{x\in K,|\alpha|\leq j} |D^\alpha \psi|$