functional analysis, distribution

Question

$T_1(\phi) = \sum^{\infty}_{n=0}\phi(n)$. $T_2(\phi) = \sum^{\infty}_{n=0} D^n \phi(n)$, show $T_1$, $T_2$ are distribution.

I know that I should show that they are linear and continuous. But I am not sure about how to show they are continuous.

Answer 1

I'll do the second one (the first one is basically the same). Choose any compact $K$ and put it inside $(-M,M)$ where $M$ is a positive integer. If $\phi\in D_K(\Bbb R)$ (smooth, compactly supported in $K$), then $D^n \phi(n) = 0$ for all $n \ge M$ (hint: $supp(D\phi) \subset supp(\phi)$). So that:

$$|\langle T_2,\phi\rangle | \le \sum_{n=0}^M \|D^n \phi\|_{\infty}$$