TITLE: Tempered distributions of finite order?
QUESTION [5 upvotes]: Is every tempered distribution  of finite order? 
It seems that yes with the definition.

REPLY [5 votes]: Yes.  Every tempered distribution is bounded with respect to one of the seminorms $\|\phi\|_{k,m} = \sum_{|\alpha|\le k} \sum_{|\beta|\le m} \sup_x |x^\alpha D^\beta \phi(x)|$, and therefore is of finite order.  In fact we can write any tempered distribution as $D^\beta g$ for some polynomially bounded continuous function $g$ and some multi-index $\beta$.