# Tempered distributions of finite order?

Is every tempered distribution of finite order?

It seems that yes with the definition.

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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 $\beta$.