TITLE: Is dirac comb a tempered distribution?
QUESTION [5 upvotes]: Prove or disprove the statement
$$ III:=\sum_{k=-\infty}^\infty\delta_k\in \mathscr{S}'(\mathbb{R}^n)$$
where $\delta_k\varphi:=\varphi(k),\,\,\forall\varphi\in\mathscr{S}(\mathbb{R}^n)$
I tried to prove linearity and I think it holds, but I am not so sure about the continouity.

REPLY [12 votes]: Yes, it is. Notice that
$$|\langle III,\varphi \rangle| = \left |\sum_{k\in \mathbb Z} \varphi(k)\right| = \left |\sum_{k\in \mathbb Z} (1 + k^2)\frac{\varphi(k)}{1 + k^2}\right|\leq \sup_{x\in \mathbb R} |(1 + x^2) \varphi(x)| \sum_{k\in \mathbb Z} \frac{1}{1 + k^2} $$
Also 
$\sum_{k\in \mathbb Z} \frac{1}{1 + k^2}<\infty$ and $\sup_{x\in \mathbb R} |(1 + x^2) \varphi(x)|$ is clearly a seminorm in the Schwartz space so $III$ is continuous.